| L(s) = 1 | − 1.46·2-s − 3-s + 0.145·4-s − 5-s + 1.46·6-s + 1.42·7-s + 2.71·8-s + 9-s + 1.46·10-s + 0.0549·11-s − 0.145·12-s + 7.07·13-s − 2.09·14-s + 15-s − 4.26·16-s + 1.64·17-s − 1.46·18-s + 1.50·19-s − 0.145·20-s − 1.42·21-s − 0.0804·22-s − 5.68·23-s − 2.71·24-s + 25-s − 10.3·26-s − 27-s + 0.207·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.577·3-s + 0.0726·4-s − 0.447·5-s + 0.597·6-s + 0.540·7-s + 0.960·8-s + 0.333·9-s + 0.463·10-s + 0.0165·11-s − 0.0419·12-s + 1.96·13-s − 0.559·14-s + 0.258·15-s − 1.06·16-s + 0.399·17-s − 0.345·18-s + 0.345·19-s − 0.0324·20-s − 0.311·21-s − 0.0171·22-s − 1.18·23-s − 0.554·24-s + 0.200·25-s − 2.03·26-s − 0.192·27-s + 0.0392·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 - 0.0549T + 11T^{2} \) |
| 13 | \( 1 - 7.07T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 + 5.68T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 - 0.146T + 31T^{2} \) |
| 37 | \( 1 - 3.10T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + 4.64T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 4.83T + 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 4.23T + 79T^{2} \) |
| 83 | \( 1 + 4.13T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−7.899214943673120444905518424268, −7.29539716472715330252544152492, −6.31378973634375587593939021379, −5.73650697646566326251080058175, −4.78613050510436438582113728944, −4.08928462145878096389319155070, −3.35839720163722911084496083420, −1.75197387418906568704529939677, −1.17554905688159916715091558434, 0,
1.17554905688159916715091558434, 1.75197387418906568704529939677, 3.35839720163722911084496083420, 4.08928462145878096389319155070, 4.78613050510436438582113728944, 5.73650697646566326251080058175, 6.31378973634375587593939021379, 7.29539716472715330252544152492, 7.899214943673120444905518424268
