| L(s) = 1 | − 7.72·3-s + 13.6·5-s − 7·7-s + 32.6·9-s − 23.9·11-s + 58.0·13-s − 105.·15-s + 25.7·17-s − 140.·19-s + 54.0·21-s + 31.0·23-s + 61.7·25-s − 43.9·27-s − 81.6·29-s − 55.8·31-s + 184.·33-s − 95.6·35-s − 5.44·37-s − 448.·39-s − 41·41-s + 376.·43-s + 446.·45-s − 165.·47-s + 49·49-s − 199.·51-s + 372.·53-s − 326.·55-s + ⋯ |
| L(s) = 1 | − 1.48·3-s + 1.22·5-s − 0.377·7-s + 1.21·9-s − 0.655·11-s + 1.23·13-s − 1.81·15-s + 0.367·17-s − 1.69·19-s + 0.561·21-s + 0.281·23-s + 0.493·25-s − 0.313·27-s − 0.523·29-s − 0.323·31-s + 0.974·33-s − 0.461·35-s − 0.0241·37-s − 1.84·39-s − 0.156·41-s + 1.33·43-s + 1.47·45-s − 0.513·47-s + 0.142·49-s − 0.546·51-s + 0.965·53-s − 0.800·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 + 7.72T + 27T^{2} \) |
| 5 | \( 1 - 13.6T + 125T^{2} \) |
| 11 | \( 1 + 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 81.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 55.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.44T + 5.06e4T^{2} \) |
| 43 | \( 1 - 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 165.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 372.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 327.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 239.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 145.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 962.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 761.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 708.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 384.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 58.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 66.9T + 9.12e5T^{2} \) |
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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
−9.203831501634901262300985006739, −8.252989246468253655332455670839, −6.96507992113633084041550971994, −6.17339961930858678555607133355, −5.82430770181805527219199500765, −5.02061718074819095904886691914, −3.87699316195534550401598202863, −2.38721851640142507292231081992, −1.23761084875631422929713760340, 0,
1.23761084875631422929713760340, 2.38721851640142507292231081992, 3.87699316195534550401598202863, 5.02061718074819095904886691914, 5.82430770181805527219199500765, 6.17339961930858678555607133355, 6.96507992113633084041550971994, 8.252989246468253655332455670839, 9.203831501634901262300985006739
