| L(s) = 1 | + 19.4·3-s + 82.2·5-s + 200.·7-s + 135.·9-s + 121·11-s + 418.·13-s + 1.60e3·15-s + 292.·17-s − 2.60e3·19-s + 3.90e3·21-s − 3.64e3·23-s + 3.64e3·25-s − 2.08e3·27-s − 13.9·29-s − 692.·31-s + 2.35e3·33-s + 1.64e4·35-s + 4.78e3·37-s + 8.14e3·39-s − 1.81e4·41-s − 1.59e4·43-s + 1.11e4·45-s − 1.05e4·47-s + 2.33e4·49-s + 5.68e3·51-s + 2.35e4·53-s + 9.95e3·55-s + ⋯ |
| L(s) = 1 | + 1.24·3-s + 1.47·5-s + 1.54·7-s + 0.559·9-s + 0.301·11-s + 0.686·13-s + 1.83·15-s + 0.245·17-s − 1.65·19-s + 1.93·21-s − 1.43·23-s + 1.16·25-s − 0.550·27-s − 0.00307·29-s − 0.129·31-s + 0.376·33-s + 2.27·35-s + 0.575·37-s + 0.857·39-s − 1.68·41-s − 1.31·43-s + 0.823·45-s − 0.696·47-s + 1.39·49-s + 0.306·51-s + 1.15·53-s + 0.443·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.487922528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.487922528\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 - 19.4T + 243T^{2} \) |
| 5 | \( 1 - 82.2T + 3.12e3T^{2} \) |
| 7 | \( 1 - 200.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 418.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 292.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.64e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 13.9T + 2.05e7T^{2} \) |
| 31 | \( 1 + 692.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.59e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.75e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.11e5T + 8.58e9T^{2} \) |
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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
−11.78374263580951575150924057629, −10.60546710592988308651516979325, −9.693882061216968535584651939428, −8.541865127790941975086532263303, −8.184167921799683615760428183299, −6.52148973862538518303442003983, −5.33816573820448082853794309052, −3.93498995180439610569072104704, −2.22518636473064861338222809968, −1.65864343952665049043025395389,
1.65864343952665049043025395389, 2.22518636473064861338222809968, 3.93498995180439610569072104704, 5.33816573820448082853794309052, 6.52148973862538518303442003983, 8.184167921799683615760428183299, 8.541865127790941975086532263303, 9.693882061216968535584651939428, 10.60546710592988308651516979325, 11.78374263580951575150924057629
