| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 1.32·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 2.64·10-s + 64.4·11-s − 12·12-s + 49.6·13-s + 14·14-s − 3.96·15-s + 16·16-s + 4.18·17-s − 18·18-s + 98.3·19-s + 5.28·20-s + 21·21-s − 128.·22-s − 23·23-s + 24·24-s − 123.·25-s − 99.3·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.118·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.0834·10-s + 1.76·11-s − 0.288·12-s + 1.06·13-s + 0.267·14-s − 0.0681·15-s + 0.250·16-s + 0.0597·17-s − 0.235·18-s + 1.18·19-s + 0.0590·20-s + 0.218·21-s − 1.24·22-s − 0.208·23-s + 0.204·24-s − 0.986·25-s − 0.749·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.440309522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440309522\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 1.32T + 125T^{2} \) |
| 11 | \( 1 - 64.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 49.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.18T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 4.52T + 2.43e4T^{2} \) |
| 31 | \( 1 - 215.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 48.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 58.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 178.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 54.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 278.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 151.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 786.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 788.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 244.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 271.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 557.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 98.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + 471.T + 9.12e5T^{2} \) |
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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
−9.627119407427603904082073346084, −8.965590641832832207783579967032, −8.043839111384720113757820592910, −6.95720469779875271527696111564, −6.35095376930905194310767294280, −5.61237842017190641172792840058, −4.18596803174123578392511538332, −3.29529649981483569414400644772, −1.64797825803971502290037249333, −0.797310489754715182153154101402,
0.797310489754715182153154101402, 1.64797825803971502290037249333, 3.29529649981483569414400644772, 4.18596803174123578392511538332, 5.61237842017190641172792840058, 6.35095376930905194310767294280, 6.95720469779875271527696111564, 8.043839111384720113757820592910, 8.965590641832832207783579967032, 9.627119407427603904082073346084
