| L(s) = 1 | − 2.08·2-s − 0.642·3-s + 2.34·4-s − 5-s + 1.33·6-s + 4.42·7-s − 0.712·8-s − 2.58·9-s + 2.08·10-s + 11-s − 1.50·12-s − 2.18·13-s − 9.22·14-s + 0.642·15-s − 3.19·16-s − 0.356·17-s + 5.39·18-s + 5.62·19-s − 2.34·20-s − 2.84·21-s − 2.08·22-s + 2.34·23-s + 0.457·24-s + 25-s + 4.55·26-s + 3.58·27-s + 10.3·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 0.370·3-s + 1.17·4-s − 0.447·5-s + 0.546·6-s + 1.67·7-s − 0.251·8-s − 0.862·9-s + 0.658·10-s + 0.301·11-s − 0.434·12-s − 0.606·13-s − 2.46·14-s + 0.165·15-s − 0.799·16-s − 0.0865·17-s + 1.27·18-s + 1.28·19-s − 0.523·20-s − 0.620·21-s − 0.444·22-s + 0.488·23-s + 0.0933·24-s + 0.200·25-s + 0.893·26-s + 0.690·27-s + 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 3 | \( 1 + 0.642T + 3T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 0.356T + 17T^{2} \) |
| 19 | \( 1 - 5.62T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 8.78T + 29T^{2} \) |
| 31 | \( 1 + 2.69T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 6.91T + 41T^{2} \) |
| 43 | \( 1 - 1.91T + 43T^{2} \) |
| 47 | \( 1 + 3.52T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 8.57T + 59T^{2} \) |
| 61 | \( 1 - 6.86T + 61T^{2} \) |
| 67 | \( 1 + 8.42T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 8.97T + 83T^{2} \) |
| 89 | \( 1 + 7.75T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.053572896880485279415607971086, −7.55920760159245024262980253261, −7.12855946438043444886357481935, −5.87549618848374098525107857415, −5.11537382494376373243715076759, −4.46640251762353212498387132243, −3.18145305068478361470982207099, −1.99836884595676412810634710697, −1.20207772543203331612692615030, 0,
1.20207772543203331612692615030, 1.99836884595676412810634710697, 3.18145305068478361470982207099, 4.46640251762353212498387132243, 5.11537382494376373243715076759, 5.87549618848374098525107857415, 7.12855946438043444886357481935, 7.55920760159245024262980253261, 8.053572896880485279415607971086
