| L(s) = 1 | − 2·5-s + 12·11-s + 12·19-s + 25-s + 4·29-s − 4·31-s − 4·41-s − 3·49-s − 24·55-s − 32·59-s − 12·61-s + 12·71-s + 24·79-s + 28·89-s − 24·95-s + 44·101-s + 20·109-s + 18·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 8·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 3.61·11-s + 2.75·19-s + 1/5·25-s + 0.742·29-s − 0.718·31-s − 0.624·41-s − 3/7·49-s − 3.23·55-s − 4.16·59-s − 1.53·61-s + 1.42·71-s + 2.70·79-s + 2.96·89-s − 2.46·95-s + 4.37·101-s + 1.91·109-s + 1.63·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.642·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.07689539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.07689539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| good | 11 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 13 | \( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 2415 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 106 T^{2} + 5183 T^{4} - 150348 T^{6} + 5183 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 2 T + 41 T^{2} - 60 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 46 T^{2} + 1399 T^{4} - 74788 T^{6} + 1399 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 2 T + 63 T^{2} - 36 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 2839 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 154 T^{2} + 12143 T^{4} - 652332 T^{6} + 12143 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 146 T^{2} + 14103 T^{4} - 884828 T^{6} + 14103 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 16 T + 113 T^{2} + 608 T^{3} + 113 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 73 | \( 1 - 298 T^{2} + 43775 T^{4} - 3982284 T^{6} + 43775 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 319 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 8719 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.22269158455393790350914564583, −3.93895573677988932253739247407, −3.83689775190262803977540072414, −3.67513301535098410227844086987, −3.65186820106578845331241954470, −3.60571634337043927841808893470, −3.46613151040677573753265618692, −3.26206146346076700682016476130, −3.17736154454198428490515862420, −3.07158807484538320285607929457, −2.94718596344429781847333774117, −2.66795467240823144768163905718, −2.44211054645320300142899831148, −2.29593133679340886196695142038, −2.06354088447491457788936659538, −1.99525130872959086556673711418, −1.71121007533927141228239653474, −1.54388480140376012667139944279, −1.49768388564283524996941479045, −1.25428320123999939157933123893, −1.03890333790404018913158555316, −0.959680054145853547646999213604, −0.77812707746309052233231597719, −0.41805010445447578540211059537, −0.28646285925087774487799971726,
0.28646285925087774487799971726, 0.41805010445447578540211059537, 0.77812707746309052233231597719, 0.959680054145853547646999213604, 1.03890333790404018913158555316, 1.25428320123999939157933123893, 1.49768388564283524996941479045, 1.54388480140376012667139944279, 1.71121007533927141228239653474, 1.99525130872959086556673711418, 2.06354088447491457788936659538, 2.29593133679340886196695142038, 2.44211054645320300142899831148, 2.66795467240823144768163905718, 2.94718596344429781847333774117, 3.07158807484538320285607929457, 3.17736154454198428490515862420, 3.26206146346076700682016476130, 3.46613151040677573753265618692, 3.60571634337043927841808893470, 3.65186820106578845331241954470, 3.67513301535098410227844086987, 3.83689775190262803977540072414, 3.93895573677988932253739247407, 4.22269158455393790350914564583
Plot not available for L-functions of degree greater than 10.
