| L(s) = 1 | + 4.14·2-s + 2.50·3-s + 9.14·4-s − 5.59·5-s + 10.3·6-s − 20.1·7-s + 4.75·8-s − 20.7·9-s − 23.1·10-s + 1.76·11-s + 22.8·12-s + 64.1·13-s − 83.5·14-s − 14.0·15-s − 53.5·16-s − 93.3·17-s − 85.8·18-s − 52.8·19-s − 51.2·20-s − 50.5·21-s + 7.30·22-s + 11.8·24-s − 93.6·25-s + 265.·26-s − 119.·27-s − 184.·28-s + 229.·29-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 0.481·3-s + 1.14·4-s − 0.500·5-s + 0.705·6-s − 1.08·7-s + 0.209·8-s − 0.768·9-s − 0.733·10-s + 0.0483·11-s + 0.550·12-s + 1.36·13-s − 1.59·14-s − 0.241·15-s − 0.836·16-s − 1.33·17-s − 1.12·18-s − 0.638·19-s − 0.572·20-s − 0.524·21-s + 0.0707·22-s + 0.101·24-s − 0.749·25-s + 2.00·26-s − 0.851·27-s − 1.24·28-s + 1.46·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 - 4.14T + 8T^{2} \) |
| 3 | \( 1 - 2.50T + 27T^{2} \) |
| 5 | \( 1 + 5.59T + 125T^{2} \) |
| 7 | \( 1 + 20.1T + 343T^{2} \) |
| 11 | \( 1 - 1.76T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 34.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 274.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 466.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 519.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 934.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 596.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 432.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 995.T + 9.12e5T^{2} \) |
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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
−10.14727643644706554521989946717, −8.825286995776919911819391296909, −8.398867011466555331530656771885, −6.68804951503684405685543111688, −6.35211377835194007298841317177, −5.17412348410069577417067373493, −3.92109673939522804061270559942, −3.42313187371796051218237312576, −2.34802211620169270016886550427, 0,
2.34802211620169270016886550427, 3.42313187371796051218237312576, 3.92109673939522804061270559942, 5.17412348410069577417067373493, 6.35211377835194007298841317177, 6.68804951503684405685543111688, 8.398867011466555331530656771885, 8.825286995776919911819391296909, 10.14727643644706554521989946717
