| L(s) = 1 | − 18.5·5-s − 7·7-s + 63.2·11-s − 16.7·13-s + 41.2·17-s − 39.1·19-s − 21.8·23-s + 219.·25-s − 138.·29-s + 95.6·31-s + 129.·35-s + 176.·37-s + 407.·41-s − 100.·43-s − 144.·47-s + 49·49-s + 409.·53-s − 1.17e3·55-s − 0.852·59-s − 407.·61-s + 310.·65-s + 9.38·67-s − 944.·71-s + 86.2·73-s − 443.·77-s − 563.·79-s − 969.·83-s + ⋯ |
| L(s) = 1 | − 1.65·5-s − 0.377·7-s + 1.73·11-s − 0.357·13-s + 0.589·17-s − 0.472·19-s − 0.197·23-s + 1.75·25-s − 0.884·29-s + 0.554·31-s + 0.627·35-s + 0.784·37-s + 1.55·41-s − 0.355·43-s − 0.447·47-s + 0.142·49-s + 1.06·53-s − 2.87·55-s − 0.00188·59-s − 0.854·61-s + 0.592·65-s + 0.0171·67-s − 1.57·71-s + 0.138·73-s − 0.655·77-s − 0.802·79-s − 1.28·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 + 18.5T + 125T^{2} \) |
| 11 | \( 1 - 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 95.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 407.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 100.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 144.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 0.852T + 2.05e5T^{2} \) |
| 61 | \( 1 + 407.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 9.38T + 3.00e5T^{2} \) |
| 71 | \( 1 + 944.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 86.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 563.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 969.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 956.T + 9.12e5T^{2} \) |
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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
−9.103820645796175699906032365664, −8.301678325252520975620474756340, −7.46554282342876344456980580459, −6.80377012557770652879916982991, −5.81860832442208572058686858194, −4.33042718833397799686435761914, −3.97389402412616857016455595181, −2.93079948583381291691668697449, −1.21272065390900776349106606300, 0,
1.21272065390900776349106606300, 2.93079948583381291691668697449, 3.97389402412616857016455595181, 4.33042718833397799686435761914, 5.81860832442208572058686858194, 6.80377012557770652879916982991, 7.46554282342876344456980580459, 8.301678325252520975620474756340, 9.103820645796175699906032365664
