| L(s) = 1 | − 5.21·2-s − 3·3-s + 19.2·4-s + 5.83·5-s + 15.6·6-s + 31.3·7-s − 58.6·8-s + 9·9-s − 30.4·10-s + 16.2·11-s − 57.7·12-s − 163.·14-s − 17.5·15-s + 152.·16-s − 54·17-s − 46.9·18-s − 66.3·19-s + 112.·20-s − 93.9·21-s − 84.9·22-s − 182.·23-s + 175.·24-s − 90.9·25-s − 27·27-s + 602.·28-s − 164.·29-s + 91.4·30-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.577·3-s + 2.40·4-s + 0.522·5-s + 1.06·6-s + 1.69·7-s − 2.59·8-s + 0.333·9-s − 0.963·10-s + 0.446·11-s − 1.38·12-s − 3.11·14-s − 0.301·15-s + 2.37·16-s − 0.770·17-s − 0.615·18-s − 0.801·19-s + 1.25·20-s − 0.976·21-s − 0.823·22-s − 1.65·23-s + 1.49·24-s − 0.727·25-s − 0.192·27-s + 4.06·28-s − 1.05·29-s + 0.556·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.21T + 8T^{2} \) |
| 5 | \( 1 - 5.83T + 125T^{2} \) |
| 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 - 16.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 55.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 514.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 265.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 468.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 852.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 165.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 479.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 574.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 66.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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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
−10.06774265330299205735218832661, −9.157727125330827233396767247335, −8.284508627326858919456355346936, −7.68097011861322036612463582570, −6.57971473894379572374298452953, −5.73984757001648859594199898853, −4.34484532019756185028087228717, −2.08253031609239192771700245046, −1.54535292071621979701498337693, 0,
1.54535292071621979701498337693, 2.08253031609239192771700245046, 4.34484532019756185028087228717, 5.73984757001648859594199898853, 6.57971473894379572374298452953, 7.68097011861322036612463582570, 8.284508627326858919456355346936, 9.157727125330827233396767247335, 10.06774265330299205735218832661
