| L(s) = 1 | + (−15.1 + 9.76i)2-s + (−3.84 + 26.7i)3-s + (82.3 − 180. i)4-s + (201. − 59.0i)5-s + (−202. − 443. i)6-s + (−1.07e3 − 1.23e3i)7-s + (180. + 1.25e3i)8-s + (−699. − 205. i)9-s + (−2.48e3 + 2.86e3i)10-s + (−809. − 520. i)11-s + (4.50e3 + 2.89e3i)12-s + (3.21e3 − 3.71e3i)13-s + (2.83e4 + 8.33e3i)14-s + (805. + 5.60e3i)15-s + (1.61e3 + 1.86e3i)16-s + (1.68e3 + 3.68e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.34 + 0.863i)2-s + (−0.0821 + 0.571i)3-s + (0.643 − 1.40i)4-s + (0.719 − 0.211i)5-s + (−0.382 − 0.838i)6-s + (−1.18 − 1.36i)7-s + (0.124 + 0.866i)8-s + (−0.319 − 0.0939i)9-s + (−0.784 + 0.905i)10-s + (−0.183 − 0.117i)11-s + (0.752 + 0.483i)12-s + (0.406 − 0.469i)13-s + (2.76 + 0.811i)14-s + (0.0616 + 0.428i)15-s + (0.0987 + 0.113i)16-s + (0.0831 + 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0981222 + 0.432685i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0981222 + 0.432685i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.84 - 26.7i)T \) |
| 23 | \( 1 + (-4.49e4 - 3.71e4i)T \) |
| good | 2 | \( 1 + (15.1 - 9.76i)T + (53.1 - 116. i)T^{2} \) |
| 5 | \( 1 + (-201. + 59.0i)T + (6.57e4 - 4.22e4i)T^{2} \) |
| 7 | \( 1 + (1.07e3 + 1.23e3i)T + (-1.17e5 + 8.15e5i)T^{2} \) |
| 11 | \( 1 + (809. + 520. i)T + (8.09e6 + 1.77e7i)T^{2} \) |
| 13 | \( 1 + (-3.21e3 + 3.71e3i)T + (-8.93e6 - 6.21e7i)T^{2} \) |
| 17 | \( 1 + (-1.68e3 - 3.68e3i)T + (-2.68e8 + 3.10e8i)T^{2} \) |
| 19 | \( 1 + (1.41e4 - 3.10e4i)T + (-5.85e8 - 6.75e8i)T^{2} \) |
| 29 | \( 1 + (5.66e4 + 1.24e5i)T + (-1.12e10 + 1.30e10i)T^{2} \) |
| 31 | \( 1 + (-3.15e4 - 2.19e5i)T + (-2.63e10 + 7.75e9i)T^{2} \) |
| 37 | \( 1 + (-2.96e4 - 8.71e3i)T + (7.98e10 + 5.13e10i)T^{2} \) |
| 41 | \( 1 + (3.97e5 - 1.16e5i)T + (1.63e11 - 1.05e11i)T^{2} \) |
| 43 | \( 1 + (5.28e4 - 3.67e5i)T + (-2.60e11 - 7.65e10i)T^{2} \) |
| 47 | \( 1 - 1.10e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-2.25e5 - 2.59e5i)T + (-1.67e11 + 1.16e12i)T^{2} \) |
| 59 | \( 1 + (-9.36e4 + 1.08e5i)T + (-3.54e11 - 2.46e12i)T^{2} \) |
| 61 | \( 1 + (3.54e5 + 2.46e6i)T + (-3.01e12 + 8.85e11i)T^{2} \) |
| 67 | \( 1 + (2.22e5 - 1.43e5i)T + (2.51e12 - 5.51e12i)T^{2} \) |
| 71 | \( 1 + (-2.64e6 + 1.70e6i)T + (3.77e12 - 8.27e12i)T^{2} \) |
| 73 | \( 1 + (2.59e6 - 5.68e6i)T + (-7.23e12 - 8.34e12i)T^{2} \) |
| 79 | \( 1 + (2.14e6 - 2.47e6i)T + (-2.73e12 - 1.90e13i)T^{2} \) |
| 83 | \( 1 + (-2.62e6 - 7.70e5i)T + (2.28e13 + 1.46e13i)T^{2} \) |
| 89 | \( 1 + (1.33e6 - 9.31e6i)T + (-4.24e13 - 1.24e13i)T^{2} \) |
| 97 | \( 1 + (7.98e6 - 2.34e6i)T + (6.79e13 - 4.36e13i)T^{2} \) |
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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
−13.89297328494870042066355149779, −12.89288661572878549240070604935, −10.72991029665594673295004409098, −10.05603779640587180813217716994, −9.339951453887035443309837438843, −8.011750307670276409672328278571, −6.76165757362474267878649486279, −5.74531039546917426740380726604, −3.64962831584378628114566476233, −1.07921333355717725008888120527,
0.28567099732097451869589793021, 2.03523725121706761612141097305, 2.84006755290204318750916348244, 5.78664141500184757827723477204, 6.98209321622510256968202144430, 8.716419585534381183978191493743, 9.264774076821600207386727793898, 10.37704342941733125419418325719, 11.58002702060984979577778803818, 12.52051386816826257805831406220
