| L(s) = 1 | + 2-s + 4-s + 3.46·5-s − 7-s + 8-s + 3.46·10-s + 5.46·11-s − 3.46·13-s − 14-s + 16-s − 7.46·17-s + 19-s + 3.46·20-s + 5.46·22-s + 5.46·23-s + 6.99·25-s − 3.46·26-s − 28-s + 6·29-s + 6.92·31-s + 32-s − 7.46·34-s − 3.46·35-s + 10·37-s + 38-s + 3.46·40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.54·5-s − 0.377·7-s + 0.353·8-s + 1.09·10-s + 1.64·11-s − 0.960·13-s − 0.267·14-s + 0.250·16-s − 1.81·17-s + 0.229·19-s + 0.774·20-s + 1.16·22-s + 1.13·23-s + 1.39·25-s − 0.679·26-s − 0.188·28-s + 1.11·29-s + 1.24·31-s + 0.176·32-s − 1.28·34-s − 0.585·35-s + 1.64·37-s + 0.162·38-s + 0.547·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.879977547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.879977547\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 9.46T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−9.285055372263357549392046008434, −8.297782014101703106788659932133, −6.78261303430059598583977675594, −6.68304401168267273880138709452, −5.98159371006717615689971220033, −4.87709161993679250703606891524, −4.40160320314858647422857199763, −3.03993034257982394543822731262, −2.31142045400819991841852134076, −1.26988506724232690547654711275,
1.26988506724232690547654711275, 2.31142045400819991841852134076, 3.03993034257982394543822731262, 4.40160320314858647422857199763, 4.87709161993679250703606891524, 5.98159371006717615689971220033, 6.68304401168267273880138709452, 6.78261303430059598583977675594, 8.297782014101703106788659932133, 9.285055372263357549392046008434
