L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 25·5-s + 36·6-s + 41.0·7-s + 64·8-s + 81·9-s + 100·10-s + 275.·11-s + 144·12-s + 966.·13-s + 164.·14-s + 225·15-s + 256·16-s − 1.58e3·17-s + 324·18-s − 361·19-s + 400·20-s + 369.·21-s + 1.10e3·22-s + 3.31e3·23-s + 576·24-s + 625·25-s + 3.86e3·26-s + 729·27-s + 656.·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.316·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.687·11-s + 0.288·12-s + 1.58·13-s + 0.223·14-s + 0.258·15-s + 0.250·16-s − 1.33·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.182·21-s + 0.485·22-s + 1.30·23-s + 0.204·24-s + 0.200·25-s + 1.12·26-s + 0.192·27-s + 0.158·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(5.859569176\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.859569176\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 + 361T \) |
good | 7 | \( 1 - 41.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 275.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 966.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.58e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.37e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.57e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.74e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.39e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.63e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.13e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.77e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.67e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 516.T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.48e4T + 8.58e9T^{2} \) |
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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
−10.01694539861692142635790910898, −8.811486972494674948930058464480, −8.458619338767945592026196165265, −6.93898839831177918188417454600, −6.43508734700627307198153337763, −5.24062831070303672787779126487, −4.21054040371901370734207969884, −3.32847303554462689746168677893, −2.11981832547680344591196684734, −1.14256439051025889019600735204,
1.14256439051025889019600735204, 2.11981832547680344591196684734, 3.32847303554462689746168677893, 4.21054040371901370734207969884, 5.24062831070303672787779126487, 6.43508734700627307198153337763, 6.93898839831177918188417454600, 8.458619338767945592026196165265, 8.811486972494674948930058464480, 10.01694539861692142635790910898