| L(s) = 1 | + (8.53 − 14.7i)2-s + (46.7 − 0.978i)3-s + (−81.6 − 141. i)4-s + (117. + 202. i)5-s + (384. − 699. i)6-s + (−171.5 + 297. i)7-s − 601.·8-s + (2.18e3 − 91.4i)9-s + 3.99e3·10-s + (3.33e3 − 5.78e3i)11-s + (−3.95e3 − 6.53e3i)12-s + (−4.94e3 − 8.57e3i)13-s + (2.92e3 + 5.06e3i)14-s + (5.67e3 + 9.37e3i)15-s + (5.31e3 − 9.20e3i)16-s − 1.67e4·17-s + ⋯ |
| L(s) = 1 | + (0.754 − 1.30i)2-s + (0.999 − 0.0209i)3-s + (−0.637 − 1.10i)4-s + (0.419 + 0.726i)5-s + (0.726 − 1.32i)6-s + (−0.188 + 0.327i)7-s − 0.415·8-s + (0.999 − 0.0418i)9-s + 1.26·10-s + (0.756 − 1.31i)11-s + (−0.660 − 1.09i)12-s + (−0.624 − 1.08i)13-s + (0.285 + 0.493i)14-s + (0.434 + 0.717i)15-s + (0.324 − 0.561i)16-s − 0.827·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.87408 - 3.28321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.87408 - 3.28321i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-46.7 + 0.978i)T \) |
| 7 | \( 1 + (171.5 - 297. i)T \) |
| good | 2 | \( 1 + (-8.53 + 14.7i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (-117. - 202. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-3.33e3 + 5.78e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (4.94e3 + 8.57e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + 1.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.98e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-2.61e4 - 4.53e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.27e5 - 2.20e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-7.56e4 - 1.31e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 5.66e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.52e4 + 2.63e4i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.82e5 - 4.89e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-7.80e4 + 1.35e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 3.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-2.06e5 - 3.58e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (5.89e5 - 1.02e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-5.80e5 - 1.00e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 3.64e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.90e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (2.66e6 - 4.61e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.96e6 + 5.13e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 - 9.78e4T + 4.42e13T^{2} \) |
| 97 | \( 1 + (2.68e6 - 4.65e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.31879191038108700306240864205, −12.16438477177205358851241356066, −10.97432712883316485833951810293, −9.992332527655170097066935349031, −8.852802180001710085472917557699, −7.13894290116044254110933761017, −5.30675937853786609486688429899, −3.37933169238087808242792152593, −2.92835581780801126289277623479, −1.36255357384240756881360490485,
1.81526123988989159044856730374, 4.05327040207265340480746625931, 4.93565990112440573594995267194, 6.74801193006197639717857714672, 7.48184479109248150799174488780, 8.985529379817370399691612125754, 9.789521848152598115764183050960, 12.11752313996752501076253029409, 13.27992436111741196923710191899, 13.90022049950359144201040475701
