| L(s) = 1 | + 2-s + 1.53·3-s + 4-s − 3.07·5-s + 1.53·6-s + 0.199·7-s + 8-s − 0.652·9-s − 3.07·10-s + 4.34·11-s + 1.53·12-s − 4.67·13-s + 0.199·14-s − 4.71·15-s + 16-s − 6.79·17-s − 0.652·18-s − 4.14·19-s − 3.07·20-s + 0.306·21-s + 4.34·22-s − 1.03·23-s + 1.53·24-s + 4.48·25-s − 4.67·26-s − 5.59·27-s + 0.199·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.884·3-s + 0.5·4-s − 1.37·5-s + 0.625·6-s + 0.0755·7-s + 0.353·8-s − 0.217·9-s − 0.973·10-s + 1.30·11-s + 0.442·12-s − 1.29·13-s + 0.0534·14-s − 1.21·15-s + 0.250·16-s − 1.64·17-s − 0.153·18-s − 0.950·19-s − 0.688·20-s + 0.0668·21-s + 0.926·22-s − 0.215·23-s + 0.312·24-s + 0.896·25-s − 0.915·26-s − 1.07·27-s + 0.0377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 - 0.199T + 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 + 1.03T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 - 0.399T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 4.19T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 - 2.39T + 97T^{2} \) |
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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
−8.305957459045777036139275092101, −7.81806052129711388069182133971, −6.79215105641235436646893171377, −6.45301917606903936088343401318, −4.94072535293219267630568224813, −4.32656425633488983569763249235, −3.71374876612968733200304825066, −2.82475627463786883512452545420, −1.93425979478684749146589362973, 0,
1.93425979478684749146589362973, 2.82475627463786883512452545420, 3.71374876612968733200304825066, 4.32656425633488983569763249235, 4.94072535293219267630568224813, 6.45301917606903936088343401318, 6.79215105641235436646893171377, 7.81806052129711388069182133971, 8.305957459045777036139275092101
