| L(s) = 1 | − 0.942·2-s + 3·3-s − 7.11·4-s + 1.29·5-s − 2.82·6-s + 21.4·7-s + 14.2·8-s + 9·9-s − 1.22·10-s + 11·11-s − 21.3·12-s + 2.72·13-s − 20.2·14-s + 3.88·15-s + 43.4·16-s + 79.3·17-s − 8.47·18-s − 127.·19-s − 9.21·20-s + 64.4·21-s − 10.3·22-s + 5.59·23-s + 42.7·24-s − 123.·25-s − 2.56·26-s + 27·27-s − 152.·28-s + ⋯ |
| L(s) = 1 | − 0.333·2-s + 0.577·3-s − 0.889·4-s + 0.115·5-s − 0.192·6-s + 1.16·7-s + 0.629·8-s + 0.333·9-s − 0.0385·10-s + 0.301·11-s − 0.513·12-s + 0.0581·13-s − 0.386·14-s + 0.0668·15-s + 0.679·16-s + 1.13·17-s − 0.111·18-s − 1.54·19-s − 0.102·20-s + 0.670·21-s − 0.100·22-s + 0.0507·23-s + 0.363·24-s − 0.986·25-s − 0.0193·26-s + 0.192·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 0.942T + 8T^{2} \) |
| 5 | \( 1 - 1.29T + 125T^{2} \) |
| 7 | \( 1 - 21.4T + 343T^{2} \) |
| 13 | \( 1 - 2.72T + 2.19e3T^{2} \) |
| 17 | \( 1 - 79.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 5.59T + 1.21e4T^{2} \) |
| 29 | \( 1 + 271.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 392.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 21.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 127.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 298.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 306.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 548.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 372.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 121.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 820.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 557.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 55.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3T + 9.12e5T^{2} \) |
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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
−8.342858791792739438185939687943, −7.909498887867114387838180608006, −7.18062766873890179584998825910, −5.83299586948416695541131931595, −5.14678345427652021950413184178, −4.15869947627513783675365838716, −3.63201288336849869710075745777, −2.07106669026156793294267420150, −1.38522811221024253752302861023, 0,
1.38522811221024253752302861023, 2.07106669026156793294267420150, 3.63201288336849869710075745777, 4.15869947627513783675365838716, 5.14678345427652021950413184178, 5.83299586948416695541131931595, 7.18062766873890179584998825910, 7.909498887867114387838180608006, 8.342858791792739438185939687943
