| L(s) = 1 | − 3.93·3-s − 4.29·5-s + 8.63·7-s − 11.5·9-s + 56.2·11-s − 81.6·13-s + 16.9·15-s + 119.·17-s − 69.7·19-s − 33.9·21-s − 8.78·23-s − 106.·25-s + 151.·27-s + 224.·29-s + 147.·31-s − 221.·33-s − 37.1·35-s − 303.·37-s + 321.·39-s + 367.·41-s − 265.·43-s + 49.4·45-s − 163.·47-s − 268.·49-s − 470.·51-s + 476.·53-s − 242.·55-s + ⋯ |
| L(s) = 1 | − 0.757·3-s − 0.384·5-s + 0.466·7-s − 0.425·9-s + 1.54·11-s − 1.74·13-s + 0.291·15-s + 1.70·17-s − 0.842·19-s − 0.353·21-s − 0.0796·23-s − 0.852·25-s + 1.08·27-s + 1.43·29-s + 0.853·31-s − 1.16·33-s − 0.179·35-s − 1.35·37-s + 1.31·39-s + 1.39·41-s − 0.943·43-s + 0.163·45-s − 0.507·47-s − 0.782·49-s − 1.29·51-s + 1.23·53-s − 0.593·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 89 | \( 1 + 89T \) |
| good | 3 | \( 1 + 3.93T + 27T^{2} \) |
| 5 | \( 1 + 4.29T + 125T^{2} \) |
| 7 | \( 1 - 8.63T + 343T^{2} \) |
| 11 | \( 1 - 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 81.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 8.78T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 367.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 265.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 163.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 476.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 586.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 367.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 877.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 535.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 761.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 398.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.01e3T + 5.71e5T^{2} \) |
| 97 | \( 1 - 432.T + 9.12e5T^{2} \) |
| 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
−9.776806129524020255169717050305, −8.682804569735566527355664789756, −7.83141803957646368363185040936, −6.88271983414610688706021551482, −6.00877745270253511996944891393, −5.03841292161685370203580610264, −4.20081512642534437738283882349, −2.88461352427630772167953648109, −1.33968666990283997927674091123, 0,
1.33968666990283997927674091123, 2.88461352427630772167953648109, 4.20081512642534437738283882349, 5.03841292161685370203580610264, 6.00877745270253511996944891393, 6.88271983414610688706021551482, 7.83141803957646368363185040936, 8.682804569735566527355664789756, 9.776806129524020255169717050305
