| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 25·5-s − 36·6-s − 34.2·7-s − 64·8-s + 81·9-s + 100·10-s + 148.·11-s + 144·12-s + 1.14e3·13-s + 137.·14-s − 225·15-s + 256·16-s + 1.86e3·17-s − 324·18-s − 361·19-s − 400·20-s − 308.·21-s − 595.·22-s + 356.·23-s − 576·24-s + 625·25-s − 4.56e3·26-s + 729·27-s − 548.·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 − 0.264·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.370·11-s + 0.288·12-s + 1.87·13-s + 0.187·14-s − 0.258·15-s + 0.250·16-s + 1.56·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.152·21-s − 0.262·22-s + 0.140·23-s − 0.204·24-s + 0.200·25-s − 1.32·26-s + 0.192·27-s − 0.132·28-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
| \(L(3)\) |
\(\approx\) |
\(2.008570321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008570321\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 34.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 148.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.14e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.86e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 356.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.82e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.04e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.87e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.31e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.92e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 559.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.22e3T + 8.58e9T^{2} \) |
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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
−9.674713067437949316982864316237, −9.124113544872450169392672744122, −8.172744123865122778553171911120, −7.59749518203413614836645327525, −6.49991243858038671613734097158, −5.56916812235438299134443442831, −3.83926148696020834458429942717, −3.32947817198922500001471701669, −1.77852955548984555289461001209, −0.78702533463434135601803253129,
0.78702533463434135601803253129, 1.77852955548984555289461001209, 3.32947817198922500001471701669, 3.83926148696020834458429942717, 5.56916812235438299134443442831, 6.49991243858038671613734097158, 7.59749518203413614836645327525, 8.172744123865122778553171911120, 9.124113544872450169392672744122, 9.674713067437949316982864316237
