| L(s) = 1 | + (−19.2 + 5.16i)2-s + (1.60 − 5.97i)3-s + (234. − 135. i)4-s + (61.4 − 272. i)5-s + 123. i·6-s + (−419. + 804. i)7-s + (−2.01e3 + 2.01e3i)8-s + (1.86e3 + 1.07e3i)9-s + (223. + 5.57e3i)10-s + (−665. − 1.15e3i)11-s + (−433. − 1.61e3i)12-s + (−3.10e3 − 3.10e3i)13-s + (3.93e3 − 1.76e4i)14-s + (−1.53e3 − 804. i)15-s + (1.11e4 − 1.92e4i)16-s + (−1.50e3 − 402. i)17-s + ⋯ |
| L(s) = 1 | + (−1.70 + 0.456i)2-s + (0.0342 − 0.127i)3-s + (1.83 − 1.05i)4-s + (0.219 − 0.975i)5-s + 0.233i·6-s + (−0.462 + 0.886i)7-s + (−1.39 + 1.39i)8-s + (0.850 + 0.491i)9-s + (0.0705 + 1.76i)10-s + (−0.150 − 0.261i)11-s + (−0.0724 − 0.270i)12-s + (−0.392 − 0.392i)13-s + (0.383 − 1.72i)14-s + (−0.117 − 0.0615i)15-s + (0.679 − 1.17i)16-s + (−0.0742 − 0.0198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.269855 - 0.356833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.269855 - 0.356833i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-61.4 + 272. i)T \) |
| 7 | \( 1 + (419. - 804. i)T \) |
| good | 2 | \( 1 + (19.2 - 5.16i)T + (110. - 64i)T^{2} \) |
| 3 | \( 1 + (-1.60 + 5.97i)T + (-1.89e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (665. + 1.15e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (3.10e3 + 3.10e3i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + (1.50e3 + 402. i)T + (3.55e8 + 2.05e8i)T^{2} \) |
| 19 | \( 1 + (-1.73e4 + 3.00e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.04e3 + 1.13e4i)T + (-2.94e9 + 1.70e9i)T^{2} \) |
| 29 | \( 1 + 2.11e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (1.85e5 - 1.07e5i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (5.72e5 - 1.53e5i)T + (8.22e10 - 4.74e10i)T^{2} \) |
| 41 | \( 1 + 6.58e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (-6.33e4 + 6.33e4i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (2.53e5 + 9.46e5i)T + (-4.38e11 + 2.53e11i)T^{2} \) |
| 53 | \( 1 + (-1.90e5 - 5.11e4i)T + (1.01e12 + 5.87e11i)T^{2} \) |
| 59 | \( 1 + (-2.07e5 - 3.59e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.97e4 + 2.87e4i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (9.18e5 - 3.42e6i)T + (-5.24e12 - 3.03e12i)T^{2} \) |
| 71 | \( 1 - 2.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.23e6 + 4.61e6i)T + (-9.56e12 - 5.52e12i)T^{2} \) |
| 79 | \( 1 + (2.08e6 + 1.20e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (2.37e6 + 2.37e6i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 + (2.15e6 - 3.73e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-4.67e6 + 4.67e6i)T - 8.07e13iT^{2} \) |
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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
−15.48166859487468800761441132508, −13.32947657589350019015940088801, −12.00349309098990315342068227092, −10.33346564397638129121247451977, −9.311580568984019460198401739373, −8.394731590721522958231011595641, −7.07393268212530567901597295687, −5.42520100289736911240994243287, −2.00286436707249330705067310787, −0.35306756805110078337286548844,
1.55505899916205364085184895499, 3.42291787295552515774009781249, 6.79389939499729854820136714389, 7.55334820278665013610909627084, 9.451490793795366347044959801244, 10.12964231519062616703430477092, 11.05749775488012229582249466402, 12.54729583494373569301942678685, 14.33341823528388982772416229402, 15.81721833060190677015556731150
