| L(s) = 1 | − 2-s + 3.03·3-s + 4-s − 1.94·5-s − 3.03·6-s − 1.79·7-s − 8-s + 6.24·9-s + 1.94·10-s + 5.47·11-s + 3.03·12-s − 0.765·13-s + 1.79·14-s − 5.91·15-s + 16-s − 7.27·17-s − 6.24·18-s + 19-s − 1.94·20-s − 5.46·21-s − 5.47·22-s + 4.90·23-s − 3.03·24-s − 1.21·25-s + 0.765·26-s + 9.84·27-s − 1.79·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.75·3-s + 0.5·4-s − 0.869·5-s − 1.24·6-s − 0.679·7-s − 0.353·8-s + 2.08·9-s + 0.615·10-s + 1.65·11-s + 0.877·12-s − 0.212·13-s + 0.480·14-s − 1.52·15-s + 0.250·16-s − 1.76·17-s − 1.47·18-s + 0.229·19-s − 0.434·20-s − 1.19·21-s − 1.16·22-s + 1.02·23-s − 0.620·24-s − 0.243·25-s + 0.150·26-s + 1.89·27-s − 0.339·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.465121975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.465121975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 3.03T + 3T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + 0.765T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 - 1.88T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 + 6.02T + 41T^{2} \) |
| 43 | \( 1 - 6.18T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 - 0.819T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 - 2.61T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 4.97T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 4.32T + 97T^{2} \) |
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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
−8.059363137808819171453625280338, −7.19460869764200106225390884436, −6.85079182549964965243607447336, −6.20421820091899928302001622942, −4.51539248360924429903804726344, −4.14922003557074026976077756073, −3.30083972610352763586605121926, −2.74973760834950666915021229776, −1.85312122329957904150569152408, −0.801807686156226503373875673545,
0.801807686156226503373875673545, 1.85312122329957904150569152408, 2.74973760834950666915021229776, 3.30083972610352763586605121926, 4.14922003557074026976077756073, 4.51539248360924429903804726344, 6.20421820091899928302001622942, 6.85079182549964965243607447336, 7.19460869764200106225390884436, 8.059363137808819171453625280338
