| L(s) = 1 | − 2-s + 4-s − 5-s − 0.467·7-s − 8-s + 10-s − 11-s − 3.57·13-s + 0.467·14-s + 16-s + 5.59·17-s + 0.0418·19-s − 20-s + 22-s − 3.98·23-s + 25-s + 3.57·26-s − 0.467·28-s + 7.57·29-s − 2.53·31-s − 32-s − 5.59·34-s + 0.467·35-s + 0.106·37-s − 0.0418·38-s + 40-s − 4.78·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.176·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.991·13-s + 0.125·14-s + 0.250·16-s + 1.35·17-s + 0.00961·19-s − 0.223·20-s + 0.213·22-s − 0.831·23-s + 0.200·25-s + 0.700·26-s − 0.0884·28-s + 1.40·29-s − 0.454·31-s − 0.176·32-s − 0.959·34-s + 0.0790·35-s + 0.0174·37-s − 0.00679·38-s + 0.158·40-s − 0.746·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 0.467T + 7T^{2} \) |
| 13 | \( 1 + 3.57T + 13T^{2} \) |
| 17 | \( 1 - 5.59T + 17T^{2} \) |
| 19 | \( 1 - 0.0418T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 - 0.106T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 + 0.184T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 - 5.18T + 61T^{2} \) |
| 67 | \( 1 - 3.39T + 67T^{2} \) |
| 71 | \( 1 - 7.38T + 71T^{2} \) |
| 73 | \( 1 + 9.11T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.48461088530429860326984702876, −6.95600170659234603937096828641, −6.13395562241772055894691117603, −5.37955684188496543166032958094, −4.67083247471822342186512672087, −3.69248581637561132972070942081, −2.97077592273771268511073117713, −2.16414264797404799242446221920, −1.05344574217133873320171293600, 0,
1.05344574217133873320171293600, 2.16414264797404799242446221920, 2.97077592273771268511073117713, 3.69248581637561132972070942081, 4.67083247471822342186512672087, 5.37955684188496543166032958094, 6.13395562241772055894691117603, 6.95600170659234603937096828641, 7.48461088530429860326984702876
