| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s − 2·11-s + 12-s − 4·14-s + 16-s − 2·17-s + 18-s − 4·21-s − 2·22-s − 23-s + 24-s + 27-s − 4·28-s − 4·29-s + 32-s − 2·33-s − 2·34-s + 36-s − 10·37-s + 6·41-s − 4·42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.872·21-s − 0.426·22-s − 0.208·23-s + 0.204·24-s + 0.192·27-s − 0.755·28-s − 0.742·29-s + 0.176·32-s − 0.348·33-s − 0.342·34-s + 1/6·36-s − 1.64·37-s + 0.937·41-s − 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.168420849840029793853910944578, −7.31310319662070416391959014907, −6.69762936587867738080522762207, −6.00729390501890278323839728611, −5.16107792495912297960628169571, −4.19620027639044251554847993934, −3.37748149946480537753655281631, −2.84403763116542600292068150831, −1.81680622419565840347383452699, 0,
1.81680622419565840347383452699, 2.84403763116542600292068150831, 3.37748149946480537753655281631, 4.19620027639044251554847993934, 5.16107792495912297960628169571, 6.00729390501890278323839728611, 6.69762936587867738080522762207, 7.31310319662070416391959014907, 8.168420849840029793853910944578
