| L(s) = 1 | + 0.504·3-s + 1.10·5-s − 0.448·7-s − 2.74·9-s + 3.86·11-s + 1.45·13-s + 0.556·15-s − 17-s + 0.393·19-s − 0.226·21-s + 3.79·23-s − 3.78·25-s − 2.89·27-s + 6.94·29-s + 3.94·31-s + 1.94·33-s − 0.495·35-s − 2.27·37-s + 0.731·39-s + 3.45·41-s − 3.09·43-s − 3.03·45-s + 2.45·47-s − 6.79·49-s − 0.504·51-s + 11.9·53-s + 4.26·55-s + ⋯ |
| L(s) = 1 | + 0.291·3-s + 0.493·5-s − 0.169·7-s − 0.915·9-s + 1.16·11-s + 0.402·13-s + 0.143·15-s − 0.242·17-s + 0.0903·19-s − 0.0493·21-s + 0.790·23-s − 0.756·25-s − 0.557·27-s + 1.28·29-s + 0.708·31-s + 0.338·33-s − 0.0837·35-s − 0.373·37-s + 0.117·39-s + 0.539·41-s − 0.472·43-s − 0.451·45-s + 0.357·47-s − 0.971·49-s − 0.0706·51-s + 1.63·53-s + 0.575·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.534063054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.534063054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 0.504T + 3T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 + 0.448T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 19 | \( 1 - 0.393T + 19T^{2} \) |
| 23 | \( 1 - 3.79T + 23T^{2} \) |
| 29 | \( 1 - 6.94T + 29T^{2} \) |
| 31 | \( 1 - 3.94T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 - 3.45T + 41T^{2} \) |
| 43 | \( 1 + 3.09T + 43T^{2} \) |
| 47 | \( 1 - 2.45T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 2.08T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 1.24T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 - 8.33T + 89T^{2} \) |
| 97 | \( 1 + 2.39T + 97T^{2} \) |
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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
−7.972127321598646711000883873032, −7.01462389408032102963687100460, −6.38586907348313618733724357388, −5.91160228119537753015869586353, −5.06800948018726133590911317744, −4.22815312194995406953604533264, −3.41622623826551280668361661207, −2.72818641029839216471701482209, −1.80763487088869223597750645683, −0.790565749252278465800597998666,
0.790565749252278465800597998666, 1.80763487088869223597750645683, 2.72818641029839216471701482209, 3.41622623826551280668361661207, 4.22815312194995406953604533264, 5.06800948018726133590911317744, 5.91160228119537753015869586353, 6.38586907348313618733724357388, 7.01462389408032102963687100460, 7.972127321598646711000883873032
