| L(s) = 1 | + 1.61·2-s + 2.61·3-s + 0.618·4-s + 4.23·6-s + 3.85·7-s − 2.23·8-s + 3.85·9-s + 0.618·11-s + 1.61·12-s − 5.47·13-s + 6.23·14-s − 4.85·16-s − 1.47·17-s + 6.23·18-s − 0.854·19-s + 10.0·21-s + 1.00·22-s − 1.85·23-s − 5.85·24-s − 8.85·26-s + 2.23·27-s + 2.38·28-s − 2.76·29-s + 2·31-s − 3.38·32-s + 1.61·33-s − 2.38·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 1.51·3-s + 0.309·4-s + 1.72·6-s + 1.45·7-s − 0.790·8-s + 1.28·9-s + 0.186·11-s + 0.467·12-s − 1.51·13-s + 1.66·14-s − 1.21·16-s − 0.357·17-s + 1.46·18-s − 0.195·19-s + 2.20·21-s + 0.213·22-s − 0.386·23-s − 1.19·24-s − 1.73·26-s + 0.430·27-s + 0.450·28-s − 0.513·29-s + 0.359·31-s − 0.597·32-s + 0.281·33-s − 0.408·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.995953824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.995953824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 + 0.145T + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−10.67585577943444065524151143213, −9.449278792868632388231070550206, −8.874112055324257117221947981445, −7.911269616822398899126357854055, −7.28459837446302796069541256770, −5.79472458623603909074870253530, −4.66178329471364183318805907592, −4.17784270671072447652505217018, −2.86366609520758182811783190204, −2.05048806991530153177882063242,
2.05048806991530153177882063242, 2.86366609520758182811783190204, 4.17784270671072447652505217018, 4.66178329471364183318805907592, 5.79472458623603909074870253530, 7.28459837446302796069541256770, 7.911269616822398899126357854055, 8.874112055324257117221947981445, 9.449278792868632388231070550206, 10.67585577943444065524151143213
