| L(s) = 1 | − 2-s + 0.304·3-s + 4-s + 5-s − 0.304·6-s + 2.10·7-s − 8-s − 2.90·9-s − 10-s − 3.97·11-s + 0.304·12-s − 13-s − 2.10·14-s + 0.304·15-s + 16-s + 1.05·17-s + 2.90·18-s − 6.16·19-s + 20-s + 0.639·21-s + 3.97·22-s + 8.36·23-s − 0.304·24-s + 25-s + 26-s − 1.79·27-s + 2.10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.175·3-s + 0.5·4-s + 0.447·5-s − 0.124·6-s + 0.794·7-s − 0.353·8-s − 0.969·9-s − 0.316·10-s − 1.19·11-s + 0.0878·12-s − 0.277·13-s − 0.562·14-s + 0.0785·15-s + 0.250·16-s + 0.254·17-s + 0.685·18-s − 1.41·19-s + 0.223·20-s + 0.139·21-s + 0.846·22-s + 1.74·23-s − 0.0620·24-s + 0.200·25-s + 0.196·26-s − 0.345·27-s + 0.397·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 0.304T + 3T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 + 3.97T + 11T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 + 6.16T + 19T^{2} \) |
| 23 | \( 1 - 8.36T + 23T^{2} \) |
| 29 | \( 1 + 1.02T + 29T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 + 4.99T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 + 2.31T + 53T^{2} \) |
| 59 | \( 1 - 0.426T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 4.65T + 71T^{2} \) |
| 73 | \( 1 - 0.0835T + 73T^{2} \) |
| 79 | \( 1 + 3.04T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.089142632888776134534541882870, −7.64646949215907567034407328405, −6.69784664695556336496098435916, −5.85238302884590391603452725928, −5.20740383982657674729149473681, −4.39898510068232992565790302998, −2.91587172675835351273459126821, −2.52311694043586345195167670448, −1.40832200695507513152045903730, 0,
1.40832200695507513152045903730, 2.52311694043586345195167670448, 2.91587172675835351273459126821, 4.39898510068232992565790302998, 5.20740383982657674729149473681, 5.85238302884590391603452725928, 6.69784664695556336496098435916, 7.64646949215907567034407328405, 8.089142632888776134534541882870
