| L(s) = 1 | − 2·2-s − 3-s + 4·4-s + 2·6-s + 18·7-s − 8·8-s − 26·9-s − 32·11-s − 4·12-s + 47·13-s − 36·14-s + 16·16-s − 20·17-s + 52·18-s + 36·19-s − 18·21-s + 64·22-s + 23·23-s + 8·24-s − 94·26-s + 53·27-s + 72·28-s − 27·29-s − 33·31-s − 32·32-s + 32·33-s + 40·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.192·3-s + 1/2·4-s + 0.136·6-s + 0.971·7-s − 0.353·8-s − 0.962·9-s − 0.877·11-s − 0.0962·12-s + 1.00·13-s − 0.687·14-s + 1/4·16-s − 0.285·17-s + 0.680·18-s + 0.434·19-s − 0.187·21-s + 0.620·22-s + 0.208·23-s + 0.0680·24-s − 0.709·26-s + 0.377·27-s + 0.485·28-s − 0.172·29-s − 0.191·31-s − 0.176·32-s + 0.168·33-s + 0.201·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 - 47 T + p^{3} T^{2} \) |
| 17 | \( 1 + 20 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 29 | \( 1 + 27 T + p^{3} T^{2} \) |
| 31 | \( 1 + 33 T + p^{3} T^{2} \) |
| 37 | \( 1 + 56 T + p^{3} T^{2} \) |
| 41 | \( 1 + 157 T + p^{3} T^{2} \) |
| 43 | \( 1 + 18 T + p^{3} T^{2} \) |
| 47 | \( 1 + 65 T + p^{3} T^{2} \) |
| 53 | \( 1 - 14 T + p^{3} T^{2} \) |
| 59 | \( 1 + 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 552 T + p^{3} T^{2} \) |
| 67 | \( 1 - 156 T + p^{3} T^{2} \) |
| 71 | \( 1 - 699 T + p^{3} T^{2} \) |
| 73 | \( 1 - 609 T + p^{3} T^{2} \) |
| 79 | \( 1 + 644 T + p^{3} T^{2} \) |
| 83 | \( 1 + 512 T + p^{3} T^{2} \) |
| 89 | \( 1 + 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 578 T + p^{3} T^{2} \) |
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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
−8.801694185185529980426259609488, −8.298282708418698250074903771455, −7.62030510933175435269731195022, −6.54957855607424123816460037942, −5.62161088786731270119855679104, −4.91806521365120331237022310974, −3.50425058643302021168851113884, −2.41704504641600854839838949231, −1.27910741156678225549380618519, 0,
1.27910741156678225549380618519, 2.41704504641600854839838949231, 3.50425058643302021168851113884, 4.91806521365120331237022310974, 5.62161088786731270119855679104, 6.54957855607424123816460037942, 7.62030510933175435269731195022, 8.298282708418698250074903771455, 8.801694185185529980426259609488
