| L(s) = 1 | − 1.22·2-s − 0.505·4-s − 0.403·5-s − 4.45·7-s + 3.06·8-s + 0.492·10-s + 2.17·11-s + 5.85·13-s + 5.44·14-s − 2.73·16-s − 7.98·17-s + 2.00·19-s + 0.203·20-s − 2.65·22-s − 0.679·23-s − 4.83·25-s − 7.15·26-s + 2.25·28-s + 5.44·29-s − 0.595·31-s − 2.78·32-s + 9.75·34-s + 1.79·35-s + 5.86·37-s − 2.45·38-s − 1.23·40-s − 0.396·41-s + ⋯ |
| L(s) = 1 | − 0.864·2-s − 0.252·4-s − 0.180·5-s − 1.68·7-s + 1.08·8-s + 0.155·10-s + 0.655·11-s + 1.62·13-s + 1.45·14-s − 0.683·16-s − 1.93·17-s + 0.459·19-s + 0.0456·20-s − 0.566·22-s − 0.141·23-s − 0.967·25-s − 1.40·26-s + 0.426·28-s + 1.01·29-s − 0.106·31-s − 0.492·32-s + 1.67·34-s + 0.303·35-s + 0.963·37-s − 0.397·38-s − 0.195·40-s − 0.0619·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 5 | \( 1 + 0.403T + 5T^{2} \) |
| 7 | \( 1 + 4.45T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 + 7.98T + 17T^{2} \) |
| 19 | \( 1 - 2.00T + 19T^{2} \) |
| 23 | \( 1 + 0.679T + 23T^{2} \) |
| 29 | \( 1 - 5.44T + 29T^{2} \) |
| 31 | \( 1 + 0.595T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 + 0.396T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 7.74T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 6.99T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−8.886452626347241242641667707119, −8.173740386018844072226680583368, −7.13381521475683704669050488654, −6.44635468390002678192903226362, −5.84566827077755475420891905124, −4.27597203218920322425605976827, −3.89591949686488188911594713252, −2.68384457657599639687084849184, −1.20989923673633197383546031297, 0,
1.20989923673633197383546031297, 2.68384457657599639687084849184, 3.89591949686488188911594713252, 4.27597203218920322425605976827, 5.84566827077755475420891905124, 6.44635468390002678192903226362, 7.13381521475683704669050488654, 8.173740386018844072226680583368, 8.886452626347241242641667707119
