| L(s) = 1 | + 1.83·2-s + 2.19·3-s + 1.36·4-s + 4.03·6-s + 0.364·7-s − 1.16·8-s + 1.83·9-s + 1.80·11-s + 3.00·12-s − 2.03·13-s + 0.668·14-s − 4.86·16-s + 7.66·17-s + 3.36·18-s + 4·19-s + 0.801·21-s + 3.30·22-s − 4·23-s − 2.56·24-s − 5·25-s − 3.72·26-s − 2.56·27-s + 0.497·28-s − 8.06·29-s − 3.27·31-s − 6.59·32-s + 3.96·33-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 1.26·3-s + 0.682·4-s + 1.64·6-s + 0.137·7-s − 0.412·8-s + 0.611·9-s + 0.543·11-s + 0.866·12-s − 0.563·13-s + 0.178·14-s − 1.21·16-s + 1.85·17-s + 0.793·18-s + 0.917·19-s + 0.174·21-s + 0.704·22-s − 0.834·23-s − 0.523·24-s − 25-s − 0.731·26-s − 0.493·27-s + 0.0939·28-s − 1.49·29-s − 0.587·31-s − 1.16·32-s + 0.689·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.711026584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.711026584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 0.364T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 0.105T + 43T^{2} \) |
| 47 | \( 1 - 0.768T + 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 0.331T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 0.364T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−11.24477389361926089418150021393, −9.659354643139211161475068573701, −9.363122693655639067008573650062, −8.022593457580810139579729364699, −7.42258696663777023058769626545, −5.97822306518092201198192349020, −5.18240044335181127201601851061, −3.80060202536937614821137323602, −3.35927650573894485968211194833, −2.05669483713524567618885650905,
2.05669483713524567618885650905, 3.35927650573894485968211194833, 3.80060202536937614821137323602, 5.18240044335181127201601851061, 5.97822306518092201198192349020, 7.42258696663777023058769626545, 8.022593457580810139579729364699, 9.363122693655639067008573650062, 9.659354643139211161475068573701, 11.24477389361926089418150021393
