| L(s) = 1 | + 2·2-s + 3-s + 4·4-s + 5·5-s + 2·6-s − 25·7-s + 8·8-s − 26·9-s + 10·10-s + 9·11-s + 4·12-s − 76·13-s − 50·14-s + 5·15-s + 16·16-s − 24·17-s − 52·18-s − 40·19-s + 20·20-s − 25·21-s + 18·22-s − 72·23-s + 8·24-s + 25·25-s − 152·26-s − 53·27-s − 100·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.447·5-s + 0.136·6-s − 1.34·7-s + 0.353·8-s − 0.962·9-s + 0.316·10-s + 0.246·11-s + 0.0962·12-s − 1.62·13-s − 0.954·14-s + 0.0860·15-s + 1/4·16-s − 0.342·17-s − 0.680·18-s − 0.482·19-s + 0.223·20-s − 0.259·21-s + 0.174·22-s − 0.652·23-s + 0.0680·24-s + 1/5·25-s − 1.14·26-s − 0.377·27-s − 0.674·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 37 | \( 1 - p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 + 25 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 76 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 60 T + p^{3} T^{2} \) |
| 31 | \( 1 - 26 T + p^{3} T^{2} \) |
| 41 | \( 1 - 267 T + p^{3} T^{2} \) |
| 43 | \( 1 + 382 T + p^{3} T^{2} \) |
| 47 | \( 1 - 267 T + p^{3} T^{2} \) |
| 53 | \( 1 - 171 T + p^{3} T^{2} \) |
| 59 | \( 1 - 396 T + p^{3} T^{2} \) |
| 61 | \( 1 + 898 T + p^{3} T^{2} \) |
| 67 | \( 1 + 676 T + p^{3} T^{2} \) |
| 71 | \( 1 + 21 T + p^{3} T^{2} \) |
| 73 | \( 1 + 691 T + p^{3} T^{2} \) |
| 79 | \( 1 + 394 T + p^{3} T^{2} \) |
| 83 | \( 1 - 309 T + p^{3} T^{2} \) |
| 89 | \( 1 + 918 T + p^{3} T^{2} \) |
| 97 | \( 1 + 766 T + p^{3} T^{2} \) |
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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
−10.42617987181888158078801336018, −9.694104063659642656775185664330, −8.788118695289241191045997287123, −7.44251464770785429034068769221, −6.45494147639985352612688981137, −5.72201684010363084374377095803, −4.48489158449784748873172943482, −3.13769075184624206104279313722, −2.32975337272063355173174635396, 0,
2.32975337272063355173174635396, 3.13769075184624206104279313722, 4.48489158449784748873172943482, 5.72201684010363084374377095803, 6.45494147639985352612688981137, 7.44251464770785429034068769221, 8.788118695289241191045997287123, 9.694104063659642656775185664330, 10.42617987181888158078801336018
