| L(s) = 1 | − 36·11-s + 24·19-s + 42·29-s − 6·31-s − 36·41-s + 12·49-s + 72·61-s + 2·64-s + 6·71-s + 24·79-s − 6·89-s − 42·101-s + 60·109-s + 702·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 10.8·11-s + 5.50·19-s + 7.79·29-s − 1.07·31-s − 5.62·41-s + 12/7·49-s + 9.21·61-s + 1/4·64-s + 0.712·71-s + 2.70·79-s − 0.635·89-s − 4.17·101-s + 5.74·109-s + 63.8·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.216230531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.216230531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - p T^{6} - 27 T^{10} + 21 T^{12} - 27 p^{2} T^{14} - p^{7} T^{18} + p^{12} T^{24} \) |
| 7 | \( 1 - 12 T^{2} + 24 T^{4} + 262 T^{6} - 1944 T^{8} + 8424 T^{10} - 38205 T^{12} + 8424 p^{2} T^{14} - 1944 p^{4} T^{16} + 262 p^{6} T^{18} + 24 p^{8} T^{20} - 12 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( ( 1 + 18 T + 135 T^{2} + 513 T^{3} + 675 T^{4} - 3033 T^{5} - 18422 T^{6} - 3033 p T^{7} + 675 p^{2} T^{8} + 513 p^{3} T^{9} + 135 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 13 | \( 1 + 18 T^{2} - 261 T^{4} - 8633 T^{6} - 40239 T^{8} + 953919 T^{10} + 19127910 T^{12} + 953919 p^{2} T^{14} - 40239 p^{4} T^{16} - 8633 p^{6} T^{18} - 261 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( 1 + 84 T^{2} + 3900 T^{4} + 127966 T^{6} + 3288456 T^{8} + 70189416 T^{10} + 1282121619 T^{12} + 70189416 p^{2} T^{14} + 3288456 p^{4} T^{16} + 127966 p^{6} T^{18} + 3900 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24} \) |
| 19 | \( ( 1 - 12 T + 51 T^{2} - 188 T^{3} + 1314 T^{4} - 5076 T^{5} + 12699 T^{6} - 5076 p T^{7} + 1314 p^{2} T^{8} - 188 p^{3} T^{9} + 51 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 23 | \( 1 - 72 T^{2} + 2628 T^{4} - 73730 T^{6} + 1554876 T^{8} - 25875612 T^{10} + 500700963 T^{12} - 25875612 p^{2} T^{14} + 1554876 p^{4} T^{16} - 73730 p^{6} T^{18} + 2628 p^{8} T^{20} - 72 p^{10} T^{22} + p^{12} T^{24} \) |
| 29 | \( ( 1 - 21 T + 207 T^{2} - 1125 T^{3} + 1890 T^{4} + 22776 T^{5} - 210023 T^{6} + 22776 p T^{7} + 1890 p^{2} T^{8} - 1125 p^{3} T^{9} + 207 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + 3 T + 24 T^{2} + 80 T^{3} + 423 T^{4} - 4653 T^{5} - 5787 T^{6} - 4653 p T^{7} + 423 p^{2} T^{8} + 80 p^{3} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 37 | \( 1 + 132 T^{2} + 8412 T^{4} + 391870 T^{6} + 16229592 T^{8} + 584807544 T^{10} + 20378076867 T^{12} + 584807544 p^{2} T^{14} + 16229592 p^{4} T^{16} + 391870 p^{6} T^{18} + 8412 p^{8} T^{20} + 132 p^{10} T^{22} + p^{12} T^{24} \) |
| 41 | \( ( 1 + 18 T + 171 T^{2} + 1143 T^{3} + 9009 T^{4} + 79119 T^{5} + 597574 T^{6} + 79119 p T^{7} + 9009 p^{2} T^{8} + 1143 p^{3} T^{9} + 171 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 43 | \( 1 - 276 T^{2} + 40134 T^{4} - 3953309 T^{6} + 292635369 T^{8} - 17126403723 T^{10} + 814017920361 T^{12} - 17126403723 p^{2} T^{14} + 292635369 p^{4} T^{16} - 3953309 p^{6} T^{18} + 40134 p^{8} T^{20} - 276 p^{10} T^{22} + p^{12} T^{24} \) |
| 47 | \( 1 + 36 T^{2} + 2574 T^{4} - 23141 T^{6} - 166887 T^{8} + 70522893 T^{10} + 7143449961 T^{12} + 70522893 p^{2} T^{14} - 166887 p^{4} T^{16} - 23141 p^{6} T^{18} + 2574 p^{8} T^{20} + 36 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( ( 1 - 84 T^{2} + 6072 T^{4} - 362545 T^{6} + 6072 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 144 T^{2} - 576 T^{3} + 144 p T^{4} - 97596 T^{5} + 406765 T^{6} - 97596 p T^{7} + 144 p^{3} T^{8} - 576 p^{3} T^{9} + 144 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 36 T + 756 T^{2} - 190 p T^{3} + 141336 T^{4} - 1424232 T^{5} + 12082719 T^{6} - 1424232 p T^{7} + 141336 p^{2} T^{8} - 190 p^{4} T^{9} + 756 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 - 144 T^{2} + 11844 T^{4} - 912746 T^{6} + 87406884 T^{8} - 6957232452 T^{10} + 499884420891 T^{12} - 6957232452 p^{2} T^{14} + 87406884 p^{4} T^{16} - 912746 p^{6} T^{18} + 11844 p^{8} T^{20} - 144 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( ( 1 - 3 T - 150 T^{2} - 63 T^{3} + 13371 T^{4} + 19698 T^{5} - 1081649 T^{6} + 19698 p T^{7} + 13371 p^{2} T^{8} - 63 p^{3} T^{9} - 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 73 | \( 1 + 408 T^{2} + 95268 T^{4} + 15399598 T^{6} + 1905199452 T^{8} + 187847914644 T^{10} + 15132874130115 T^{12} + 187847914644 p^{2} T^{14} + 1905199452 p^{4} T^{16} + 15399598 p^{6} T^{18} + 95268 p^{8} T^{20} + 408 p^{10} T^{22} + p^{12} T^{24} \) |
| 79 | \( ( 1 - 12 T + 204 T^{2} - 2915 T^{3} + 31545 T^{4} - 326493 T^{5} + 3330513 T^{6} - 326493 p T^{7} + 31545 p^{2} T^{8} - 2915 p^{3} T^{9} + 204 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 + 468 T^{2} + 93294 T^{4} + 9694123 T^{6} + 419139873 T^{8} - 21991579731 T^{10} - 4161640619895 T^{12} - 21991579731 p^{2} T^{14} + 419139873 p^{4} T^{16} + 9694123 p^{6} T^{18} + 93294 p^{8} T^{20} + 468 p^{10} T^{22} + p^{12} T^{24} \) |
| 89 | \( ( 1 + 3 T - 186 T^{2} - 585 T^{3} + 18915 T^{4} + 33708 T^{5} - 1703351 T^{6} + 33708 p T^{7} + 18915 p^{2} T^{8} - 585 p^{3} T^{9} - 186 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 97 | \( 1 + 93 T^{2} - 1770 T^{4} - 1760936 T^{6} - 103886055 T^{8} + 10655050167 T^{10} + 2102618450697 T^{12} + 10655050167 p^{2} T^{14} - 103886055 p^{4} T^{16} - 1760936 p^{6} T^{18} - 1770 p^{8} T^{20} + 93 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.33252427444800281087361549331, −3.32583475421166310092946574270, −3.25350824375078114235048697771, −3.05328623206781612081605159449, −2.97443540934827751570052927838, −2.85449468057580580454719898934, −2.70252703236316582122298059914, −2.66085796499277951586985337821, −2.59296822880133799517056746840, −2.50679298021494174889355704822, −2.45189780804244914999831040983, −2.40108414581629192260220702580, −2.32169170507917870786312608845, −2.24135412528041975636787624262, −2.08399801724437977880282913809, −1.79262514607155529390763129522, −1.71934796596379884188946755897, −1.48445222708352362642519633307, −1.21192609207470900952189099355, −1.14658906272257958365919149968, −0.77684381566177173291517508519, −0.62849322601942629842673597837, −0.61989409660249548682358214669, −0.49223401117379135420759553065, −0.38994917036326733227311862101,
0.38994917036326733227311862101, 0.49223401117379135420759553065, 0.61989409660249548682358214669, 0.62849322601942629842673597837, 0.77684381566177173291517508519, 1.14658906272257958365919149968, 1.21192609207470900952189099355, 1.48445222708352362642519633307, 1.71934796596379884188946755897, 1.79262514607155529390763129522, 2.08399801724437977880282913809, 2.24135412528041975636787624262, 2.32169170507917870786312608845, 2.40108414581629192260220702580, 2.45189780804244914999831040983, 2.50679298021494174889355704822, 2.59296822880133799517056746840, 2.66085796499277951586985337821, 2.70252703236316582122298059914, 2.85449468057580580454719898934, 2.97443540934827751570052927838, 3.05328623206781612081605159449, 3.25350824375078114235048697771, 3.32583475421166310092946574270, 3.33252427444800281087361549331
Plot not available for L-functions of degree greater than 10.
