| L(s) = 1 | − 3.11·3-s − 0.609·5-s + 6.72·9-s + 5.72·11-s + 6.84·13-s + 1.90·15-s − 3.72·17-s + 19-s − 5.39·23-s − 4.62·25-s − 11.6·27-s − 3.72·29-s + 1.72·31-s − 17.8·33-s − 1.82·37-s − 21.3·39-s + 9.96·41-s − 1.21·43-s − 4.09·45-s − 11.6·47-s + 11.6·51-s + 6.88·53-s − 3.49·55-s − 3.11·57-s − 12.8·59-s − 8.60·61-s − 4.17·65-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 0.272·5-s + 2.24·9-s + 1.72·11-s + 1.89·13-s + 0.490·15-s − 0.904·17-s + 0.229·19-s − 1.12·23-s − 0.925·25-s − 2.23·27-s − 0.692·29-s + 0.310·31-s − 3.11·33-s − 0.300·37-s − 3.41·39-s + 1.55·41-s − 0.185·43-s − 0.611·45-s − 1.69·47-s + 1.62·51-s + 0.945·53-s − 0.470·55-s − 0.413·57-s − 1.67·59-s − 1.10·61-s − 0.517·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 + 0.609T + 5T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 - 6.84T + 13T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 23 | \( 1 + 5.39T + 23T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 6.88T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 8.60T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 + 2.88T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 4.41T + 97T^{2} \) |
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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
−7.33984942645514261648787406881, −6.45504260117696873720999270318, −6.20886946278756331033184711050, −5.75471474987667985202094133174, −4.62832055760423489223581036792, −4.09185718769196604326508871058, −3.56343973981937643559862048343, −1.75009613149689886915631288277, −1.16822334494967776324007369386, 0,
1.16822334494967776324007369386, 1.75009613149689886915631288277, 3.56343973981937643559862048343, 4.09185718769196604326508871058, 4.62832055760423489223581036792, 5.75471474987667985202094133174, 6.20886946278756331033184711050, 6.45504260117696873720999270318, 7.33984942645514261648787406881
