| L(s) = 1 | + 2-s + 4-s + 1.58·5-s + 8-s + 1.58·10-s + 1.58·11-s + 4.81·13-s + 16-s + 5.39·17-s − 7.09·19-s + 1.58·20-s + 1.58·22-s − 0.300·23-s − 2.47·25-s + 4.81·26-s + 8.27·29-s + 2.71·31-s + 32-s + 5.39·34-s − 37-s − 7.09·38-s + 1.58·40-s − 5.87·41-s + 1.66·43-s + 1.58·44-s − 0.300·46-s + 2.66·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.710·5-s + 0.353·8-s + 0.502·10-s + 0.478·11-s + 1.33·13-s + 0.250·16-s + 1.30·17-s − 1.62·19-s + 0.355·20-s + 0.338·22-s − 0.0626·23-s − 0.495·25-s + 0.943·26-s + 1.53·29-s + 0.487·31-s + 0.176·32-s + 0.925·34-s − 0.164·37-s − 1.15·38-s + 0.251·40-s − 0.917·41-s + 0.254·43-s + 0.239·44-s − 0.0442·46-s + 0.388·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.529148247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.529148247\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 - 5.39T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 + 0.300T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 - 4.88T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 8.38T + 79T^{2} \) |
| 83 | \( 1 + 2.36T + 83T^{2} \) |
| 89 | \( 1 + 3.21T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−7.88339697352180824075490622520, −6.81924148953008984967112266524, −6.32462031773942956104074743655, −5.85018927830208220303362116432, −5.10667756409909393017682259675, −4.19706864257875474098804322510, −3.64998068092732394163873289486, −2.74388257622433649285428951609, −1.84705407200841790236038005094, −1.01468622779629239786804554514,
1.01468622779629239786804554514, 1.84705407200841790236038005094, 2.74388257622433649285428951609, 3.64998068092732394163873289486, 4.19706864257875474098804322510, 5.10667756409909393017682259675, 5.85018927830208220303362116432, 6.32462031773942956104074743655, 6.81924148953008984967112266524, 7.88339697352180824075490622520
