L(s) = 1 | + 6·13-s + 18·19-s + 72·25-s + 18·31-s + 96·37-s − 18·43-s − 102·49-s − 90·61-s − 126·67-s + 246·73-s + 126·79-s − 150·97-s − 258·103-s + 54·109-s + 312·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 801·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 6/13·13-s + 0.947·19-s + 2.87·25-s + 0.580·31-s + 2.59·37-s − 0.418·43-s − 2.08·49-s − 1.47·61-s − 1.88·67-s + 3.36·73-s + 1.59·79-s − 1.54·97-s − 2.50·103-s + 0.495·109-s + 2.57·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4.73·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(7.766937617\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.766937617\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 72 T^{2} + 3507 T^{4} - 101808 T^{6} + 3507 p^{4} T^{8} - 72 p^{8} T^{10} + p^{12} T^{12} \) |
| 7 | \( ( 1 + 51 T^{2} + 288 T^{3} + 51 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 11 | \( 1 - 312 T^{2} + 57459 T^{4} - 8298768 T^{6} + 57459 p^{4} T^{8} - 312 p^{8} T^{10} + p^{12} T^{12} \) |
| 13 | \( ( 1 - 3 T + 414 T^{2} - 1207 T^{3} + 414 p^{2} T^{4} - 3 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 17 | \( 1 - 1656 T^{2} + 1164579 T^{4} - 444765456 T^{6} + 1164579 p^{4} T^{8} - 1656 p^{8} T^{10} + p^{12} T^{12} \) |
| 19 | \( ( 1 - 9 T - 42 T^{2} + 163 T^{3} - 42 p^{2} T^{4} - 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 - 1128 T^{2} + 1156803 T^{4} - 652681968 T^{6} + 1156803 p^{4} T^{8} - 1128 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( 1 + 234 T^{2} + 763839 T^{4} - 336354356 T^{6} + 763839 p^{4} T^{8} + 234 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( ( 1 - 9 T + 1182 T^{2} - 1235 p T^{3} + 1182 p^{2} T^{4} - 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 37 | \( ( 1 - 48 T + 3723 T^{2} - 113856 T^{3} + 3723 p^{2} T^{4} - 48 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( 1 - 5094 T^{2} + 14718447 T^{4} - 30152306900 T^{6} + 14718447 p^{4} T^{8} - 5094 p^{8} T^{10} + p^{12} T^{12} \) |
| 43 | \( ( 1 + 9 T + 3846 T^{2} + 12509 T^{3} + 3846 p^{2} T^{4} + 9 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( 1 - 5670 T^{2} + 8846415 T^{4} - 5066767316 T^{6} + 8846415 p^{4} T^{8} - 5670 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 13512 T^{2} + 82937715 T^{4} - 297035491632 T^{6} + 82937715 p^{4} T^{8} - 13512 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( 1 - 16728 T^{2} + 128508339 T^{4} - 572240181072 T^{6} + 128508339 p^{4} T^{8} - 16728 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 + 45 T + 6654 T^{2} + 333881 T^{3} + 6654 p^{2} T^{4} + 45 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( ( 1 + 63 T + 8742 T^{2} + 305371 T^{3} + 8742 p^{2} T^{4} + 63 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 11688 T^{2} + 94972035 T^{4} - 560320238064 T^{6} + 94972035 p^{4} T^{8} - 11688 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 123 T + 19686 T^{2} - 1306735 T^{3} + 19686 p^{2} T^{4} - 123 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( ( 1 - 63 T + 10830 T^{2} - 567019 T^{3} + 10830 p^{2} T^{4} - 63 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 83 | \( 1 - 7320 T^{2} + 21226515 T^{4} - 158664178896 T^{6} + 21226515 p^{4} T^{8} - 7320 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( 1 - 27654 T^{2} + 337636527 T^{4} - 2866913516180 T^{6} + 337636527 p^{4} T^{8} - 27654 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( ( 1 + 75 T + 23958 T^{2} + 1420831 T^{3} + 23958 p^{2} T^{4} + 75 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−4.61887401160163734460550027668, −4.61532505373057755581213969447, −4.45659270090801229599763509708, −3.92340158149617158382395778415, −3.90003259016227448635777701448, −3.86095365719207733995971601901, −3.78864981677355625360349976100, −3.46394172140105899017111975398, −3.34112954708306935057857965089, −3.09902341663684914285978586651, −2.95088890973523611670623858257, −2.84863839380990615280886672927, −2.67918282894603970137383388499, −2.61124090329347938578545868169, −2.42215302565646168744017603121, −2.06571845341390616229178959469, −1.99169650897195487406410808661, −1.63649603015007466447459845565, −1.42568312329051131185530845183, −1.23771609920614457617640530990, −1.18595499387294095401509938849, −0.929639283127780784997494667179, −0.72300641974786999997290684432, −0.37364621889676064734229794508, −0.24833266305792052613228581947,
0.24833266305792052613228581947, 0.37364621889676064734229794508, 0.72300641974786999997290684432, 0.929639283127780784997494667179, 1.18595499387294095401509938849, 1.23771609920614457617640530990, 1.42568312329051131185530845183, 1.63649603015007466447459845565, 1.99169650897195487406410808661, 2.06571845341390616229178959469, 2.42215302565646168744017603121, 2.61124090329347938578545868169, 2.67918282894603970137383388499, 2.84863839380990615280886672927, 2.95088890973523611670623858257, 3.09902341663684914285978586651, 3.34112954708306935057857965089, 3.46394172140105899017111975398, 3.78864981677355625360349976100, 3.86095365719207733995971601901, 3.90003259016227448635777701448, 3.92340158149617158382395778415, 4.45659270090801229599763509708, 4.61532505373057755581213969447, 4.61887401160163734460550027668
Plot not available for L-functions of degree greater than 10.