| L(s) = 1 | − 2.17·2-s − 3-s + 2.72·4-s + 5-s + 2.17·6-s − 3.13·7-s − 1.58·8-s + 9-s − 2.17·10-s + 1.65·11-s − 2.72·12-s + 0.116·13-s + 6.80·14-s − 15-s − 2.01·16-s + 5.85·17-s − 2.17·18-s − 2.97·19-s + 2.72·20-s + 3.13·21-s − 3.60·22-s + 1.98·23-s + 1.58·24-s + 25-s − 0.253·26-s − 27-s − 8.54·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.577·3-s + 1.36·4-s + 0.447·5-s + 0.887·6-s − 1.18·7-s − 0.559·8-s + 0.333·9-s − 0.687·10-s + 0.500·11-s − 0.787·12-s + 0.0323·13-s + 1.81·14-s − 0.258·15-s − 0.503·16-s + 1.41·17-s − 0.512·18-s − 0.681·19-s + 0.609·20-s + 0.683·21-s − 0.769·22-s + 0.413·23-s + 0.322·24-s + 0.200·25-s − 0.0497·26-s − 0.192·27-s − 1.61·28-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 | \( 1 + 2.17T + 2T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 - 0.116T + 13T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 + 9.73T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 0.844T + 37T^{2} \) |
| 41 | \( 1 + 8.67T + 41T^{2} \) |
| 43 | \( 1 - 0.156T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 0.975T + 53T^{2} \) |
| 59 | \( 1 - 2.78T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 5.57T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 6.74T + 83T^{2} \) |
| 89 | \( 1 - 1.86T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.64045466889658585149214500284, −7.21561214048530649525707793064, −6.35207181859273343461481714622, −6.00178873892007715090866955422, −5.02281297650593395754636873417, −3.89991348112671528087939998086, −3.02295465570121936558472075871, −1.91509648419332604895851720902, −1.03557797683764147543099230623, 0,
1.03557797683764147543099230623, 1.91509648419332604895851720902, 3.02295465570121936558472075871, 3.89991348112671528087939998086, 5.02281297650593395754636873417, 6.00178873892007715090866955422, 6.35207181859273343461481714622, 7.21561214048530649525707793064, 7.64045466889658585149214500284
