L(s) = 1 | − 3·3-s − 5·5-s + 23.2·7-s + 9·9-s + 49.6·11-s − 13·13-s + 15·15-s − 17.0·17-s − 129.·19-s − 69.6·21-s − 149.·23-s + 25·25-s − 27·27-s + 11.3·29-s − 75.4·31-s − 148.·33-s − 116.·35-s + 84.5·37-s + 39·39-s + 136.·41-s − 240.·43-s − 45·45-s + 5.99·47-s + 196.·49-s + 51.1·51-s + 25.9·53-s − 248.·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.25·7-s + 0.333·9-s + 1.36·11-s − 0.277·13-s + 0.258·15-s − 0.243·17-s − 1.55·19-s − 0.723·21-s − 1.35·23-s + 0.200·25-s − 0.192·27-s + 0.0725·29-s − 0.437·31-s − 0.785·33-s − 0.560·35-s + 0.375·37-s + 0.160·39-s + 0.518·41-s − 0.854·43-s − 0.149·45-s + 0.0185·47-s + 0.571·49-s + 0.140·51-s + 0.0673·53-s − 0.608·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
good | 7 | \( 1 - 23.2T + 343T^{2} \) |
| 11 | \( 1 - 49.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 17.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 11.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 75.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 240.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 5.99T + 1.03e5T^{2} \) |
| 53 | \( 1 - 25.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 386.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 476.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 298.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 663.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 99.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 480.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 397.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 194.T + 9.12e5T^{2} \) |
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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
−8.518676119977125003815950192380, −7.988990592694534793606883726736, −6.96172647867344658739160258685, −6.30247643373145379031681826543, −5.31979655437724279948806287512, −4.32519990976052328912840829037, −3.95253382604192390448869248035, −2.21322549307073211445083538119, −1.32097446149692544758585933941, 0,
1.32097446149692544758585933941, 2.21322549307073211445083538119, 3.95253382604192390448869248035, 4.32519990976052328912840829037, 5.31979655437724279948806287512, 6.30247643373145379031681826543, 6.96172647867344658739160258685, 7.988990592694534793606883726736, 8.518676119977125003815950192380