| L(s) = 1 | + 7.65·5-s + 31.5·7-s − 7.18·11-s + 84.3·13-s − 17·17-s + 37.0·19-s + 150.·23-s − 66.3·25-s + 11.5·29-s + 53.2·31-s + 241.·35-s − 99.2·37-s − 118.·41-s + 456.·43-s + 571.·47-s + 654.·49-s − 462.·53-s − 55.0·55-s + 48.0·59-s + 59.5·61-s + 645.·65-s + 740.·67-s − 930.·71-s − 697.·73-s − 227.·77-s − 1.03e3·79-s − 22.2·83-s + ⋯ |
| L(s) = 1 | + 0.684·5-s + 1.70·7-s − 0.197·11-s + 1.79·13-s − 0.242·17-s + 0.447·19-s + 1.36·23-s − 0.531·25-s + 0.0741·29-s + 0.308·31-s + 1.16·35-s − 0.440·37-s − 0.450·41-s + 1.61·43-s + 1.77·47-s + 1.90·49-s − 1.19·53-s − 0.134·55-s + 0.106·59-s + 0.124·61-s + 1.23·65-s + 1.35·67-s − 1.55·71-s − 1.11·73-s − 0.336·77-s − 1.47·79-s − 0.0293·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.111908557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.111908557\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 5 | \( 1 - 7.65T + 125T^{2} \) |
| 7 | \( 1 - 31.5T + 343T^{2} \) |
| 11 | \( 1 + 7.18T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 53.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 456.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 571.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 48.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 59.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 740.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 697.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 22.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 369.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−8.700965848353450004636584915674, −7.88717242997736034082683398344, −7.16092731771886564816007469334, −6.07571218220993549198776733205, −5.51212991965518512515293461359, −4.69916424495866465772388139561, −3.84710206520990420013856502611, −2.65232681989972167573277539641, −1.60573875120427005624182431351, −1.01731806861660403315051617288,
1.01731806861660403315051617288, 1.60573875120427005624182431351, 2.65232681989972167573277539641, 3.84710206520990420013856502611, 4.69916424495866465772388139561, 5.51212991965518512515293461359, 6.07571218220993549198776733205, 7.16092731771886564816007469334, 7.88717242997736034082683398344, 8.700965848353450004636584915674
