| L(s) = 1 | − 1.67·3-s + 1.05·5-s + 1.62·7-s − 0.179·9-s + 2.87·11-s + 13-s − 1.77·15-s + 4.55·17-s + 7.96·19-s − 2.73·21-s + 4.44·23-s − 3.88·25-s + 5.33·27-s − 29-s + 5.89·31-s − 4.83·33-s + 1.72·35-s − 5.01·37-s − 1.67·39-s + 10.8·41-s + 2.63·43-s − 0.189·45-s − 3.52·47-s − 4.34·49-s − 7.64·51-s + 0.493·53-s + 3.04·55-s + ⋯ |
| L(s) = 1 | − 0.969·3-s + 0.472·5-s + 0.615·7-s − 0.0597·9-s + 0.867·11-s + 0.277·13-s − 0.458·15-s + 1.10·17-s + 1.82·19-s − 0.596·21-s + 0.926·23-s − 0.776·25-s + 1.02·27-s − 0.185·29-s + 1.05·31-s − 0.841·33-s + 0.290·35-s − 0.823·37-s − 0.268·39-s + 1.68·41-s + 0.402·43-s − 0.0282·45-s − 0.513·47-s − 0.621·49-s − 1.07·51-s + 0.0678·53-s + 0.410·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.019611939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.019611939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 - 1.05T + 5T^{2} \) |
| 7 | \( 1 - 1.62T + 7T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 17 | \( 1 - 4.55T + 17T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 23 | \( 1 - 4.44T + 23T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 2.63T + 43T^{2} \) |
| 47 | \( 1 + 3.52T + 47T^{2} \) |
| 53 | \( 1 - 0.493T + 53T^{2} \) |
| 59 | \( 1 + 3.14T + 59T^{2} \) |
| 61 | \( 1 + 7.35T + 61T^{2} \) |
| 67 | \( 1 + 8.26T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 2.43T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 6.35T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.85750908071129712986484305537, −7.44147087032462070294973169136, −6.39098003891758476058757578356, −5.96350372023536402746902122290, −5.24290536999730497098438971767, −4.75236510739843592271184582976, −3.63233596684960471727862662220, −2.84886846725409014129617617006, −1.49490662147377805926723337073, −0.887084715520281318181415636793,
0.887084715520281318181415636793, 1.49490662147377805926723337073, 2.84886846725409014129617617006, 3.63233596684960471727862662220, 4.75236510739843592271184582976, 5.24290536999730497098438971767, 5.96350372023536402746902122290, 6.39098003891758476058757578356, 7.44147087032462070294973169136, 7.85750908071129712986484305537
