| L(s) = 1 | + 0.236·2-s + 3·3-s − 7.94·4-s − 5·5-s + 0.708·6-s − 7·7-s − 3.76·8-s + 9·9-s − 1.18·10-s − 50.4·11-s − 23.8·12-s − 80.9·13-s − 1.65·14-s − 15·15-s + 62.6·16-s + 76.3·17-s + 2.12·18-s + 4.13·19-s + 39.7·20-s − 21·21-s − 11.9·22-s − 204.·23-s − 11.2·24-s + 25·25-s − 19.1·26-s + 27·27-s + 55.6·28-s + ⋯ |
| L(s) = 1 | + 0.0834·2-s + 0.577·3-s − 0.993·4-s − 0.447·5-s + 0.0481·6-s − 0.377·7-s − 0.166·8-s + 0.333·9-s − 0.0373·10-s − 1.38·11-s − 0.573·12-s − 1.72·13-s − 0.0315·14-s − 0.258·15-s + 0.979·16-s + 1.08·17-s + 0.0278·18-s + 0.0499·19-s + 0.444·20-s − 0.218·21-s − 0.115·22-s − 1.85·23-s − 0.0960·24-s + 0.200·25-s − 0.144·26-s + 0.192·27-s + 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 0.236T + 8T^{2} \) |
| 11 | \( 1 + 50.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 80.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.13T + 6.85e3T^{2} \) |
| 23 | \( 1 + 204.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 91.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 198.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 155.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 175.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 312.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 154.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 734.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 678.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 60.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 116.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 916.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−12.78558300354075809399407693352, −12.09629816446006541908408907175, −10.16767996106242022361660541788, −9.684910701443008864898681857926, −8.182929901094436683442313395985, −7.56005056030982890340253285064, −5.51723347212477855927934184572, −4.29591832719651791160000479572, −2.81449756126596726250098903133, 0,
2.81449756126596726250098903133, 4.29591832719651791160000479572, 5.51723347212477855927934184572, 7.56005056030982890340253285064, 8.182929901094436683442313395985, 9.684910701443008864898681857926, 10.16767996106242022361660541788, 12.09629816446006541908408907175, 12.78558300354075809399407693352
