| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s + 9.32·7-s + 8·8-s + 9·9-s − 10·10-s − 9.42·11-s − 12·12-s − 61.7·13-s + 18.6·14-s + 15·15-s + 16·16-s + 79.7·17-s + 18·18-s − 28.8·19-s − 20·20-s − 27.9·21-s − 18.8·22-s + 98.5·23-s − 24·24-s + 25·25-s − 123.·26-s − 27·27-s + 37.3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.503·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.258·11-s − 0.288·12-s − 1.31·13-s + 0.356·14-s + 0.258·15-s + 0.250·16-s + 1.13·17-s + 0.235·18-s − 0.347·19-s − 0.223·20-s − 0.290·21-s − 0.182·22-s + 0.893·23-s − 0.204·24-s + 0.200·25-s − 0.931·26-s − 0.192·27-s + 0.251·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 31 | \( 1 + 31T \) |
| good | 7 | \( 1 - 9.32T + 343T^{2} \) |
| 11 | \( 1 + 9.42T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 79.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 159.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 384.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 185.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 517.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 128.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 235.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 542.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 226.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 911.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 833.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 825.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 656.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.35e3T + 9.12e5T^{2} \) |
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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
−9.433270826614959777386553829690, −8.142324760519021992348497965922, −7.42557528497255418719036188795, −6.70492809328320395683147270246, −5.41378401212891959264626568112, −5.03747045127966607434159034286, −3.97960367146883364747428675846, −2.87268530307500238561227772563, −1.54701802672621310420871659063, 0,
1.54701802672621310420871659063, 2.87268530307500238561227772563, 3.97960367146883364747428675846, 5.03747045127966607434159034286, 5.41378401212891959264626568112, 6.70492809328320395683147270246, 7.42557528497255418719036188795, 8.142324760519021992348497965922, 9.433270826614959777386553829690
