| L(s) = 1 | − 1.61·2-s − 0.381·3-s + 0.618·4-s + 0.618·6-s − 3·7-s + 2.23·8-s − 2.85·9-s − 1.61·11-s − 0.236·12-s + 13-s + 4.85·14-s − 4.85·16-s − 0.763·17-s + 4.61·18-s + 1.14·21-s + 2.61·22-s − 5.38·23-s − 0.854·24-s − 1.61·26-s + 2.23·27-s − 1.85·28-s − 3.61·29-s − 8.85·31-s + 3.38·32-s + 0.618·33-s + 1.23·34-s − 1.76·36-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.220·3-s + 0.309·4-s + 0.252·6-s − 1.13·7-s + 0.790·8-s − 0.951·9-s − 0.487·11-s − 0.0681·12-s + 0.277·13-s + 1.29·14-s − 1.21·16-s − 0.185·17-s + 1.08·18-s + 0.250·21-s + 0.558·22-s − 1.12·23-s − 0.174·24-s − 0.317·26-s + 0.430·27-s − 0.350·28-s − 0.671·29-s − 1.59·31-s + 0.597·32-s + 0.107·33-s + 0.211·34-s − 0.293·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 + 8.85T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 0.145T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 0.326T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 + 7.76T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.10291334505861014976348259278, −6.58108024432842240100928909302, −5.69495890711204515545351744512, −5.23212234624897575972634101903, −4.06612607677978245676459272664, −3.40283148983620936945769696008, −2.45509737793981009181570837059, −1.54156526200735277543915975528, 0, 0,
1.54156526200735277543915975528, 2.45509737793981009181570837059, 3.40283148983620936945769696008, 4.06612607677978245676459272664, 5.23212234624897575972634101903, 5.69495890711204515545351744512, 6.58108024432842240100928909302, 7.10291334505861014976348259278
