| L(s) = 1 | + (1.19 − 0.871i)2-s + (−38.8 + 119. i)4-s + (−171. − 124. i)5-s + (−463. + 1.42e3i)7-s + (116. + 357. i)8-s − 313.·10-s + (2.44e3 + 3.67e3i)11-s + (3.99e3 − 2.90e3i)13-s + (686. + 2.11e3i)14-s + (−1.25e4 − 9.13e3i)16-s + (−1.84e4 − 1.34e4i)17-s + (7.99e3 + 2.46e4i)19-s + (2.15e4 − 1.56e4i)20-s + (6.13e3 + 2.27e3i)22-s − 3.90e4·23-s + ⋯ |
| L(s) = 1 | + (0.105 − 0.0770i)2-s + (−0.303 + 0.934i)4-s + (−0.612 − 0.444i)5-s + (−0.510 + 1.57i)7-s + (0.0802 + 0.247i)8-s − 0.0991·10-s + (0.554 + 0.832i)11-s + (0.504 − 0.366i)13-s + (0.0668 + 0.205i)14-s + (−0.767 − 0.557i)16-s + (−0.910 − 0.661i)17-s + (0.267 + 0.823i)19-s + (0.601 − 0.437i)20-s + (0.122 + 0.0455i)22-s − 0.668·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0629579 - 0.134118i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0629579 - 0.134118i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-2.44e3 - 3.67e3i)T \) |
| good | 2 | \( 1 + (-1.19 + 0.871i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (171. + 124. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (463. - 1.42e3i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-3.99e3 + 2.90e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.84e4 + 1.34e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-7.99e3 - 2.46e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 3.90e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (1.00e4 - 3.07e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-1.58e5 + 1.15e5i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-3.61e3 + 1.11e4i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.80e5 + 5.54e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 6.01e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.86e5 - 5.74e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.00e6 - 7.26e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-9.28e5 + 2.85e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.90e6 + 1.38e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 1.91e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-1.92e6 - 1.39e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-6.06e5 + 1.86e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (2.81e6 - 2.04e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-2.38e5 - 1.73e5i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 - 2.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-5.27e6 + 3.83e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.79866296185371074004226557906, −12.20047294702229630676546323251, −11.56151295899257196735130144267, −9.638195039725316304171665863279, −8.746091807274292328922490477380, −7.897626629637632972286342392413, −6.37913968997760583591109976883, −4.86640974457319540205633182461, −3.63753268745123710542941683621, −2.27175206932089275251691416249,
0.04879168070246663706390579065, 1.21695084724568847996206927889, 3.53654276077165710536262355150, 4.46553808430346048931139533096, 6.27510357039095059587791835377, 6.99085062597279857232048054562, 8.530537055473873190505212605923, 9.835881671776030225735918590243, 10.76937721730458411668988328471, 11.48368302861449104965492036709
