| L(s) = 1 | − 2-s − 7·4-s + 5·5-s + 7·7-s + 15·8-s − 5·10-s − 12·11-s − 78·13-s − 7·14-s + 41·16-s + 94·17-s + 40·19-s − 35·20-s + 12·22-s − 32·23-s + 25·25-s + 78·26-s − 49·28-s + 50·29-s − 248·31-s − 161·32-s − 94·34-s + 35·35-s − 434·37-s − 40·38-s + 75·40-s − 402·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s + 0.447·5-s + 0.377·7-s + 0.662·8-s − 0.158·10-s − 0.328·11-s − 1.66·13-s − 0.133·14-s + 0.640·16-s + 1.34·17-s + 0.482·19-s − 0.391·20-s + 0.116·22-s − 0.290·23-s + 1/5·25-s + 0.588·26-s − 0.330·28-s + 0.320·29-s − 1.43·31-s − 0.889·32-s − 0.474·34-s + 0.169·35-s − 1.92·37-s − 0.170·38-s + 0.296·40-s − 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 32 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 434 T + p^{3} T^{2} \) |
| 41 | \( 1 + 402 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 + 536 T + p^{3} T^{2} \) |
| 53 | \( 1 + 22 T + p^{3} T^{2} \) |
| 59 | \( 1 - 560 T + p^{3} T^{2} \) |
| 61 | \( 1 + 278 T + p^{3} T^{2} \) |
| 67 | \( 1 + 164 T + p^{3} T^{2} \) |
| 71 | \( 1 + 672 T + p^{3} T^{2} \) |
| 73 | \( 1 - 82 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1000 T + p^{3} T^{2} \) |
| 83 | \( 1 - 448 T + p^{3} T^{2} \) |
| 89 | \( 1 - 870 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1026 T + p^{3} T^{2} \) |
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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
−10.29093452296666508814143357731, −9.953707418086912419155153838378, −8.953483597381048389930069261851, −7.945959594022488307530582268645, −7.14627912529197374497683297216, −5.42039604938517925824522052087, −4.92184649284991937487056158766, −3.36382697788484203649707309246, −1.68474003499304263991997865840, 0,
1.68474003499304263991997865840, 3.36382697788484203649707309246, 4.92184649284991937487056158766, 5.42039604938517925824522052087, 7.14627912529197374497683297216, 7.945959594022488307530582268645, 8.953483597381048389930069261851, 9.953707418086912419155153838378, 10.29093452296666508814143357731
