| L(s) = 1 | − 0.428·3-s + 5-s − 4.67·7-s − 2.81·9-s + 2.67·11-s + 2.24·13-s − 0.428·15-s + 2·17-s + 19-s + 2·21-s + 0.672·23-s + 25-s + 2.48·27-s − 1.14·29-s − 1.14·31-s − 1.14·33-s − 4.67·35-s − 4.24·37-s − 0.960·39-s + 4.85·41-s − 5.52·43-s − 2.81·45-s − 4.67·47-s + 14.8·49-s − 0.856·51-s + 0.244·53-s + 2.67·55-s + ⋯ |
| L(s) = 1 | − 0.247·3-s + 0.447·5-s − 1.76·7-s − 0.938·9-s + 0.805·11-s + 0.622·13-s − 0.110·15-s + 0.485·17-s + 0.229·19-s + 0.436·21-s + 0.140·23-s + 0.200·25-s + 0.479·27-s − 0.212·29-s − 0.205·31-s − 0.199·33-s − 0.789·35-s − 0.697·37-s − 0.153·39-s + 0.758·41-s − 0.843·43-s − 0.419·45-s − 0.681·47-s + 2.11·49-s − 0.119·51-s + 0.0336·53-s + 0.360·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.428T + 3T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 - 0.672T + 23T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 + 1.14T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 4.85T + 41T^{2} \) |
| 43 | \( 1 + 5.52T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 - 0.244T + 53T^{2} \) |
| 59 | \( 1 - 0.287T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 9.20T + 67T^{2} \) |
| 71 | \( 1 + 9.14T + 71T^{2} \) |
| 73 | \( 1 - 1.14T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 0.183T + 83T^{2} \) |
| 89 | \( 1 - 9.14T + 89T^{2} \) |
| 97 | \( 1 - 9.10T + 97T^{2} \) |
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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
−7.60572773535254502965514619010, −6.77032996011102969807019121939, −6.23644400771493316606274566123, −5.82626470158477895417442188207, −4.99139999841812444166902904393, −3.73388024444353338280112966497, −3.33259525588562444012583992444, −2.47586530452988934410423216348, −1.18249750225369321640110377319, 0,
1.18249750225369321640110377319, 2.47586530452988934410423216348, 3.33259525588562444012583992444, 3.73388024444353338280112966497, 4.99139999841812444166902904393, 5.82626470158477895417442188207, 6.23644400771493316606274566123, 6.77032996011102969807019121939, 7.60572773535254502965514619010
