| L(s) = 1 | − 0.321·3-s + 3.25·5-s + 3.32·7-s − 2.89·9-s + 2.53·11-s − 2.04·13-s − 1.04·15-s + 17-s + 2.05·19-s − 1.06·21-s + 7.93·23-s + 5.61·25-s + 1.89·27-s + 1.07·29-s + 8.38·31-s − 0.815·33-s + 10.8·35-s − 7.00·37-s + 0.657·39-s + 9.45·41-s − 12.5·43-s − 9.43·45-s + 4.27·47-s + 4.06·49-s − 0.321·51-s − 12.2·53-s + 8.26·55-s + ⋯ |
| L(s) = 1 | − 0.185·3-s + 1.45·5-s + 1.25·7-s − 0.965·9-s + 0.764·11-s − 0.567·13-s − 0.270·15-s + 0.242·17-s + 0.471·19-s − 0.233·21-s + 1.65·23-s + 1.12·25-s + 0.364·27-s + 0.198·29-s + 1.50·31-s − 0.141·33-s + 1.83·35-s − 1.15·37-s + 0.105·39-s + 1.47·41-s − 1.91·43-s − 1.40·45-s + 0.622·47-s + 0.580·49-s − 0.0449·51-s − 1.67·53-s + 1.11·55-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
| \(L(1)\) |
\(\approx\) |
\(3.267853254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.267853254\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.321T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 + 2.04T + 13T^{2} \) |
| 19 | \( 1 - 2.05T + 19T^{2} \) |
| 23 | \( 1 - 7.93T + 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 - 8.38T + 31T^{2} \) |
| 37 | \( 1 + 7.00T + 37T^{2} \) |
| 41 | \( 1 - 9.45T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 4.27T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 + 7.10T + 67T^{2} \) |
| 71 | \( 1 - 5.06T + 71T^{2} \) |
| 73 | \( 1 - 2.15T + 73T^{2} \) |
| 79 | \( 1 + 1.27T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 5.13T + 97T^{2} \) |
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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
−7.895804135702735756548359738026, −7.02186652672300488431997113817, −6.36125332311832641987977063708, −5.71115881703020494370787312141, −4.99845038220813868179885385376, −4.71865854240001090239696110674, −3.30679596340042247322010234525, −2.57730846649662955245610812735, −1.71391621376567789349996216590, −0.975083077784675653430596809995,
0.975083077784675653430596809995, 1.71391621376567789349996216590, 2.57730846649662955245610812735, 3.30679596340042247322010234525, 4.71865854240001090239696110674, 4.99845038220813868179885385376, 5.71115881703020494370787312141, 6.36125332311832641987977063708, 7.02186652672300488431997113817, 7.895804135702735756548359738026
