| L(s) = 1 | + 5.52·3-s − 68.9·7-s − 212.·9-s + 486.·11-s − 428.·13-s + 1.80e3·17-s + 1.04e3·19-s − 380.·21-s − 686.·23-s − 2.51e3·27-s − 1.33e3·29-s − 7.99e3·31-s + 2.68e3·33-s − 1.97e3·37-s − 2.36e3·39-s + 1.07e4·41-s − 1.50e4·43-s − 895.·47-s − 1.20e4·49-s + 9.94e3·51-s − 1.93e4·53-s + 5.78e3·57-s − 2.11e4·59-s − 2.77e4·61-s + 1.46e4·63-s + 7.71e3·67-s − 3.79e3·69-s + ⋯ |
| L(s) = 1 | + 0.354·3-s − 0.531·7-s − 0.874·9-s + 1.21·11-s − 0.703·13-s + 1.51·17-s + 0.665·19-s − 0.188·21-s − 0.270·23-s − 0.664·27-s − 0.295·29-s − 1.49·31-s + 0.429·33-s − 0.236·37-s − 0.249·39-s + 1.00·41-s − 1.23·43-s − 0.0591·47-s − 0.717·49-s + 0.535·51-s − 0.945·53-s + 0.235·57-s − 0.792·59-s − 0.953·61-s + 0.465·63-s + 0.210·67-s − 0.0959·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.52T + 243T^{2} \) |
| 7 | \( 1 + 68.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 486.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 428.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 686.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 895.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.71e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.22e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.48e5T + 8.58e9T^{2} \) |
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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
−9.663760769952137382378764424603, −9.326389073826344049558828735669, −8.143048867060283140636510069865, −7.26543367833479084050704529738, −6.15034666101399144060676425011, −5.24612615091120063156159555819, −3.74478506992973619234538897316, −2.98033014974405254573320178382, −1.50412646747505076647988632968, 0,
1.50412646747505076647988632968, 2.98033014974405254573320178382, 3.74478506992973619234538897316, 5.24612615091120063156159555819, 6.15034666101399144060676425011, 7.26543367833479084050704529738, 8.143048867060283140636510069865, 9.326389073826344049558828735669, 9.663760769952137382378764424603
