| L(s) = 1 | − 8·9-s + 10·11-s + 18·23-s − 10·25-s + 6·29-s + 22·37-s + 18·43-s + 18·53-s + 42·67-s + 70·71-s + 6·79-s + 25·81-s − 80·99-s + 34·107-s − 34·109-s − 34·113-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 66·169-s + ⋯ |
| L(s) = 1 | − 8/3·9-s + 3.01·11-s + 3.75·23-s − 2·25-s + 1.11·29-s + 3.61·37-s + 2.74·43-s + 2.47·53-s + 5.13·67-s + 8.30·71-s + 0.675·79-s + 25/9·81-s − 8.04·99-s + 3.28·107-s − 3.25·109-s − 3.19·113-s − 0.272·121-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 − 5.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.99744928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.99744928\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 8 T^{2} + 13 p T^{4} + 136 T^{6} + 13 p^{3} T^{8} + 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 2 p T^{2} + 43 T^{4} + 172 T^{6} + 43 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 5 T + 39 T^{2} - 111 T^{3} + 39 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 66 T^{2} + 1931 T^{4} + 32284 T^{6} + 1931 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 4 p T^{2} + 2063 T^{4} + 40656 T^{6} + 2063 p^{2} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 62 T^{2} + 2075 T^{4} + 46452 T^{6} + 2075 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 9 T + 89 T^{2} - 413 T^{3} + 89 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 3 T + 41 T^{2} - 77 T^{3} + 41 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 102 T^{2} + 5763 T^{4} + 220452 T^{6} + 5763 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 11 T + 121 T^{2} - 785 T^{3} + 121 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 68 T^{2} + 4783 T^{4} + 172400 T^{6} + 4783 p^{2} T^{8} + 68 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 9 T + 107 T^{2} - 703 T^{3} + 107 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 102 T^{2} + 8723 T^{4} + 447940 T^{6} + 8723 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 9 T + 137 T^{2} - 785 T^{3} + 137 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 104 T^{2} + 7319 T^{4} + 321864 T^{6} + 7319 p^{2} T^{8} + 104 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{3} \) |
| 67 | \( ( 1 - 21 T + 327 T^{2} - 3003 T^{3} + 327 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 - 35 T + 605 T^{2} - 6349 T^{3} + 605 p T^{4} - 35 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 160 T^{2} + 16755 T^{4} - 1307968 T^{6} + 16755 p^{2} T^{8} - 160 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 3 T + 191 T^{2} - 377 T^{3} + 191 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 94 T^{2} + 12907 T^{4} + 1359316 T^{6} + 12907 p^{2} T^{8} + 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 272 T^{2} + 42899 T^{4} + 4696608 T^{6} + 42899 p^{2} T^{8} + 272 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 156 T^{2} + 6239 T^{4} - 161984 T^{6} + 6239 p^{2} T^{8} + 156 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.29947248251644297645542804663, −4.97103439994820299319535900659, −4.62789734904495232310055880556, −4.57086243315683524382290419663, −4.38543875638402465666438387929, −4.15516136819730195616206554458, −4.02487222749628480877460871885, −3.78992273020447975579273851059, −3.78477323484975233790279753817, −3.70822966161787314479386544860, −3.40898688538079370097719369606, −3.25938192070564262017549896925, −3.22056843918062837648030010800, −2.60652924774461467992502161592, −2.53174468656135967231162991089, −2.51485471534760412396797865373, −2.36368958617679596055206082747, −2.33668314519768920898049135987, −2.18558702243811896530820776559, −1.36633536197692454933587495219, −1.32550265073589511338380538925, −1.13782917304358289711655046381, −0.799227404715373828270647488053, −0.68012540842431608522374918118, −0.58540457290451395878214399452,
0.58540457290451395878214399452, 0.68012540842431608522374918118, 0.799227404715373828270647488053, 1.13782917304358289711655046381, 1.32550265073589511338380538925, 1.36633536197692454933587495219, 2.18558702243811896530820776559, 2.33668314519768920898049135987, 2.36368958617679596055206082747, 2.51485471534760412396797865373, 2.53174468656135967231162991089, 2.60652924774461467992502161592, 3.22056843918062837648030010800, 3.25938192070564262017549896925, 3.40898688538079370097719369606, 3.70822966161787314479386544860, 3.78477323484975233790279753817, 3.78992273020447975579273851059, 4.02487222749628480877460871885, 4.15516136819730195616206554458, 4.38543875638402465666438387929, 4.57086243315683524382290419663, 4.62789734904495232310055880556, 4.97103439994820299319535900659, 5.29947248251644297645542804663
Plot not available for L-functions of degree greater than 10.
