| L(s) = 1 | − 2-s + 2.20·3-s + 4-s − 1.98·5-s − 2.20·6-s + 7-s − 8-s + 1.85·9-s + 1.98·10-s + 3.26·11-s + 2.20·12-s + 3.91·13-s − 14-s − 4.38·15-s + 16-s + 0.415·17-s − 1.85·18-s + 2.32·19-s − 1.98·20-s + 2.20·21-s − 3.26·22-s + 3.79·23-s − 2.20·24-s − 1.04·25-s − 3.91·26-s − 2.52·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.27·3-s + 0.5·4-s − 0.889·5-s − 0.899·6-s + 0.377·7-s − 0.353·8-s + 0.618·9-s + 0.628·10-s + 0.984·11-s + 0.636·12-s + 1.08·13-s − 0.267·14-s − 1.13·15-s + 0.250·16-s + 0.100·17-s − 0.437·18-s + 0.533·19-s − 0.444·20-s + 0.480·21-s − 0.696·22-s + 0.790·23-s − 0.449·24-s − 0.208·25-s − 0.768·26-s − 0.485·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.334226429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.334226429\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.20T + 3T^{2} \) |
| 5 | \( 1 + 1.98T + 5T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 - 0.415T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 - 3.79T + 23T^{2} \) |
| 29 | \( 1 - 1.25T + 29T^{2} \) |
| 31 | \( 1 + 2.01T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 - 1.01T + 41T^{2} \) |
| 43 | \( 1 - 5.84T + 43T^{2} \) |
| 47 | \( 1 + 1.31T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 + 2.06T + 61T^{2} \) |
| 67 | \( 1 + 6.66T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 5.42T + 73T^{2} \) |
| 79 | \( 1 - 9.73T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.093791032482164399960437864823, −7.68220079210684170340827001828, −6.97857661446251860851665039433, −6.19092926983310787847867656600, −5.17358830509000640211962303911, −3.93987503429795552325007904614, −3.67618698671281753506942074264, −2.79005736218721897858317376116, −1.77686168604199255859023310972, −0.882504000985937198643129403340,
0.882504000985937198643129403340, 1.77686168604199255859023310972, 2.79005736218721897858317376116, 3.67618698671281753506942074264, 3.93987503429795552325007904614, 5.17358830509000640211962303911, 6.19092926983310787847867656600, 6.97857661446251860851665039433, 7.68220079210684170340827001828, 8.093791032482164399960437864823
