| L(s) = 1 | + 2-s + 4-s − 5-s − 1.26·7-s + 8-s − 10-s − 1.46·11-s + 1.46·13-s − 1.26·14-s + 16-s + 1.46·17-s − 4.19·19-s − 20-s − 1.46·22-s + 8·23-s + 25-s + 1.46·26-s − 1.26·28-s + 8.92·29-s − 2.73·31-s + 32-s + 1.46·34-s + 1.26·35-s + 37-s − 4.19·38-s − 40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.479·7-s + 0.353·8-s − 0.316·10-s − 0.441·11-s + 0.406·13-s − 0.338·14-s + 0.250·16-s + 0.355·17-s − 0.962·19-s − 0.223·20-s − 0.312·22-s + 1.66·23-s + 0.200·25-s + 0.287·26-s − 0.239·28-s + 1.65·29-s − 0.490·31-s + 0.176·32-s + 0.251·34-s + 0.214·35-s + 0.164·37-s − 0.680·38-s − 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.550952866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550952866\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 0.196T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 5.26T + 79T^{2} \) |
| 83 | \( 1 - 5.26T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.451375181889865509583324277715, −7.918902895858996860740536914729, −6.75584917677428637725725806710, −6.61529116519936591247028677725, −5.42330427898711946486797224180, −4.84498498111578633097509312256, −3.88609616880719151677548468068, −3.18383948621967498861679021003, −2.31923939303283907845607874635, −0.859662100440955635595558502502,
0.859662100440955635595558502502, 2.31923939303283907845607874635, 3.18383948621967498861679021003, 3.88609616880719151677548468068, 4.84498498111578633097509312256, 5.42330427898711946486797224180, 6.61529116519936591247028677725, 6.75584917677428637725725806710, 7.918902895858996860740536914729, 8.451375181889865509583324277715
