| L(s) = 1 | + (−16.6 + 12.1i)2-s + (−25.4 + 78.2i)3-s + (91.6 − 282. i)4-s + (120. − 252. i)5-s + (−523. − 1.61e3i)6-s + 289.·7-s + (1.07e3 + 3.30e3i)8-s + (−3.70e3 − 2.68e3i)9-s + (1.05e3 + 5.66e3i)10-s + (3.92e3 − 2.85e3i)11-s + (1.97e4 + 1.43e4i)12-s + (−7.54e3 − 5.48e3i)13-s + (−4.81e3 + 3.50e3i)14-s + (1.66e4 + 1.58e4i)15-s + (−2.71e4 − 1.97e4i)16-s + (−3.53e3 − 1.08e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 + 1.07i)2-s + (−0.543 + 1.67i)3-s + (0.715 − 2.20i)4-s + (0.429 − 0.903i)5-s + (−0.989 − 3.04i)6-s + 0.318·7-s + (0.741 + 2.28i)8-s + (−1.69 − 1.22i)9-s + (0.333 + 1.79i)10-s + (0.889 − 0.646i)11-s + (3.29 + 2.39i)12-s + (−0.952 − 0.691i)13-s + (−0.469 + 0.341i)14-s + (1.27 + 1.20i)15-s + (−1.65 − 1.20i)16-s + (−0.174 − 0.537i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.530442 + 0.133902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.530442 + 0.133902i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-120. + 252. i)T \) |
| good | 2 | \( 1 + (16.6 - 12.1i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (25.4 - 78.2i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 7 | \( 1 - 289.T + 8.23e5T^{2} \) |
| 11 | \( 1 + (-3.92e3 + 2.85e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (7.54e3 + 5.48e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (3.53e3 + 1.08e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-5.12e3 - 1.57e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-1.85e4 + 1.35e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-7.12e4 + 2.19e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-2.71e4 - 8.36e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.54e5 - 1.12e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-4.96e5 - 3.60e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 3.33e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.69e5 + 8.30e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.97e5 + 6.07e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.27e6 + 9.28e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-7.90e5 + 5.74e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-5.01e5 - 1.54e6i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-7.95e4 + 2.44e5i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.04e6 + 7.57e5i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-9.72e5 + 2.99e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (8.52e5 + 2.62e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (1.18e6 - 8.62e5i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-1.24e6 + 3.83e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−16.43806865931749650121057562847, −15.45530206768670626803645452346, −14.39675654499250716508948435754, −11.54894329181835158182090641341, −10.10837672440410235373866881880, −9.436939596748361370196256474637, −8.312917767802546684491119577651, −6.04237105880969523629976871177, −4.86607622176454945970956346528, −0.53888034353052481609785351450,
1.34931135646341193553934237480, 2.41059660272280572048866818706, 6.73056148989857442288548415305, 7.55140396129395834699531260215, 9.264144025553035858612617947169, 10.84984370620113645085778156306, 11.77459426563504637581003598400, 12.70402627939662627225779516095, 14.31226105998826489702604415675, 16.99090595089828193043125259753
