| L(s) = 1 | − 2.84·2-s − 3·3-s + 0.104·4-s − 9.28·5-s + 8.54·6-s + 6.39·7-s + 22.4·8-s + 9·9-s + 26.4·10-s + 11·11-s − 0.313·12-s − 1.06·13-s − 18.2·14-s + 27.8·15-s − 64.8·16-s − 2.24·17-s − 25.6·18-s + 21.5·19-s − 0.971·20-s − 19.1·21-s − 31.3·22-s + 83.1·23-s − 67.4·24-s − 38.7·25-s + 3.02·26-s − 27·27-s + 0.668·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.577·3-s + 0.0130·4-s − 0.830·5-s + 0.581·6-s + 0.345·7-s + 0.993·8-s + 0.333·9-s + 0.836·10-s + 0.301·11-s − 0.00754·12-s − 0.0226·13-s − 0.347·14-s + 0.479·15-s − 1.01·16-s − 0.0320·17-s − 0.335·18-s + 0.259·19-s − 0.0108·20-s − 0.199·21-s − 0.303·22-s + 0.753·23-s − 0.573·24-s − 0.309·25-s + 0.0227·26-s − 0.192·27-s + 0.00451·28-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 | \( 1 + 2.84T + 8T^{2} \) |
| 5 | \( 1 + 9.28T + 125T^{2} \) |
| 7 | \( 1 - 6.39T + 343T^{2} \) |
| 13 | \( 1 + 1.06T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.24T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 83.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 16.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 17.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 279.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 565.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 388.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 20.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 358.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 400.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 355.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 312.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.45T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.373614658446437631612101408870, −7.63965406035272519450574106747, −7.24607668137068784170169656854, −6.12368266864367020705863018089, −5.10984715997024529080315256372, −4.35369884869967228958247306037, −3.52552946010469539388792939140, −1.93455556805127956177199369778, −0.909212622172620562788475649234, 0,
0.909212622172620562788475649234, 1.93455556805127956177199369778, 3.52552946010469539388792939140, 4.35369884869967228958247306037, 5.10984715997024529080315256372, 6.12368266864367020705863018089, 7.24607668137068784170169656854, 7.63965406035272519450574106747, 8.373614658446437631612101408870
