| L(s) = 1 | + 21.2·3-s + 45.9·5-s − 213.·7-s + 208.·9-s + 697.·11-s + 1.03e3·13-s + 976.·15-s − 967.·17-s + 1.98e3·19-s − 4.54e3·21-s + 1.86e3·23-s − 1.01e3·25-s − 738.·27-s + 4.57e3·29-s − 961·31-s + 1.48e4·33-s − 9.83e3·35-s + 6.41e3·37-s + 2.18e4·39-s − 1.75e4·41-s + 2.02e3·43-s + 9.57e3·45-s − 4.40e3·47-s + 2.89e4·49-s − 2.05e4·51-s − 2.37e4·53-s + 3.20e4·55-s + ⋯ |
| L(s) = 1 | + 1.36·3-s + 0.822·5-s − 1.64·7-s + 0.856·9-s + 1.73·11-s + 1.69·13-s + 1.12·15-s − 0.812·17-s + 1.25·19-s − 2.24·21-s + 0.735·23-s − 0.323·25-s − 0.194·27-s + 1.01·29-s − 0.179·31-s + 2.36·33-s − 1.35·35-s + 0.769·37-s + 2.30·39-s − 1.62·41-s + 0.167·43-s + 0.704·45-s − 0.290·47-s + 1.72·49-s − 1.10·51-s − 1.16·53-s + 1.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.339772976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.339772976\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + 961T \) |
| good | 3 | \( 1 - 21.2T + 243T^{2} \) |
| 5 | \( 1 - 45.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 213.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 697.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 967.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.98e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.86e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.57e3T + 2.05e7T^{2} \) |
| 37 | \( 1 - 6.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.75e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.02e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.13e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.75e5T + 8.58e9T^{2} \) |
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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
−12.92647818220880005211278413282, −11.46970872096781320887981802452, −9.872011084532555437024239461443, −9.273256140693639986326240918262, −8.596118035311162673210990566899, −6.82976382358641230824115955597, −6.10648165391422828085434815360, −3.81081984581924638861940636832, −3.01764757730796250501303734012, −1.38054276073517071387131202870,
1.38054276073517071387131202870, 3.01764757730796250501303734012, 3.81081984581924638861940636832, 6.10648165391422828085434815360, 6.82976382358641230824115955597, 8.596118035311162673210990566899, 9.273256140693639986326240918262, 9.872011084532555437024239461443, 11.46970872096781320887981802452, 12.92647818220880005211278413282
