| L(s) = 1 | + 3·2-s + 4-s − 9·5-s − 15·7-s − 21·8-s − 27·10-s − 48·11-s − 45·14-s − 71·16-s − 45·17-s − 6·19-s − 9·20-s − 144·22-s + 162·23-s − 44·25-s − 15·28-s + 144·29-s − 264·31-s − 45·32-s − 135·34-s + 135·35-s − 303·37-s − 18·38-s + 189·40-s − 192·41-s + 97·43-s − 48·44-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s − 0.804·5-s − 0.809·7-s − 0.928·8-s − 0.853·10-s − 1.31·11-s − 0.859·14-s − 1.10·16-s − 0.642·17-s − 0.0724·19-s − 0.100·20-s − 1.39·22-s + 1.46·23-s − 0.351·25-s − 0.101·28-s + 0.922·29-s − 1.52·31-s − 0.248·32-s − 0.680·34-s + 0.651·35-s − 1.34·37-s − 0.0768·38-s + 0.747·40-s − 0.731·41-s + 0.344·43-s − 0.164·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.018932255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018932255\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 15 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 17 | \( 1 + 45 T + p^{3} T^{2} \) |
| 19 | \( 1 + 6 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 - 144 T + p^{3} T^{2} \) |
| 31 | \( 1 + 264 T + p^{3} T^{2} \) |
| 37 | \( 1 + 303 T + p^{3} T^{2} \) |
| 41 | \( 1 + 192 T + p^{3} T^{2} \) |
| 43 | \( 1 - 97 T + p^{3} T^{2} \) |
| 47 | \( 1 - 111 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 522 T + p^{3} T^{2} \) |
| 61 | \( 1 - 376 T + p^{3} T^{2} \) |
| 67 | \( 1 - 36 T + p^{3} T^{2} \) |
| 71 | \( 1 - 357 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + 830 T + p^{3} T^{2} \) |
| 83 | \( 1 + 438 T + p^{3} T^{2} \) |
| 89 | \( 1 + 438 T + p^{3} T^{2} \) |
| 97 | \( 1 - 852 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.976942760629447238606902445799, −8.358419432408291463467998485874, −7.25173457937375322870501830821, −6.64425485580142140233175617013, −5.49947688925844799235584599437, −4.99110435950193780490023977481, −3.94948274356930410424839514114, −3.29056901060642994485092905087, −2.41951943457579296897232703824, −0.39243248543384265385656516099,
0.39243248543384265385656516099, 2.41951943457579296897232703824, 3.29056901060642994485092905087, 3.94948274356930410424839514114, 4.99110435950193780490023977481, 5.49947688925844799235584599437, 6.64425485580142140233175617013, 7.25173457937375322870501830821, 8.358419432408291463467998485874, 8.976942760629447238606902445799
