| L(s) = 1 | + 4.53·2-s + 12.5·4-s + 5·5-s − 2.63·7-s + 20.6·8-s + 22.6·10-s + 20.9·11-s + 60.9·13-s − 11.9·14-s − 6.70·16-s + 86.8·17-s + 41.8·19-s + 62.8·20-s + 94.9·22-s − 97.3·23-s + 25·25-s + 276.·26-s − 33.1·28-s + 157.·29-s + 95.3·31-s − 195.·32-s + 393.·34-s − 13.1·35-s − 160.·37-s + 189.·38-s + 103.·40-s − 233.·41-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.57·4-s + 0.447·5-s − 0.142·7-s + 0.914·8-s + 0.716·10-s + 0.573·11-s + 1.30·13-s − 0.228·14-s − 0.104·16-s + 1.23·17-s + 0.504·19-s + 0.702·20-s + 0.919·22-s − 0.882·23-s + 0.200·25-s + 2.08·26-s − 0.223·28-s + 1.00·29-s + 0.552·31-s − 1.08·32-s + 1.98·34-s − 0.0636·35-s − 0.714·37-s + 0.809·38-s + 0.408·40-s − 0.888·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.484147751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.484147751\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 4.53T + 8T^{2} \) |
| 7 | \( 1 + 2.63T + 343T^{2} \) |
| 11 | \( 1 - 20.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 86.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 95.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 233.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 487.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 24.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 709.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 191.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 744.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 823.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 132.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 704.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 401.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 530.T + 9.12e5T^{2} \) |
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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
−11.14242411111191998268967691493, −10.14063759810411469689327706231, −9.059347636984376057356777650738, −7.84039265151837454453267951900, −6.48422760622777241313078182476, −6.00091238998690395262060734209, −4.97582804973352783504304121231, −3.83768801395848739064503393112, −3.01233442448953888937411138180, −1.44180193016615173309263097663,
1.44180193016615173309263097663, 3.01233442448953888937411138180, 3.83768801395848739064503393112, 4.97582804973352783504304121231, 6.00091238998690395262060734209, 6.48422760622777241313078182476, 7.84039265151837454453267951900, 9.059347636984376057356777650738, 10.14063759810411469689327706231, 11.14242411111191998268967691493
