| L(s) = 1 | + (14.1 − 8.17i)2-s + (69.6 − 120. i)4-s − 3.43i·5-s + (1.24e3 + 720. i)7-s − 184. i·8-s + (−28.0 − 48.6i)10-s + (−4.29e3 + 2.48e3i)11-s + (2.81e3 + 7.40e3i)13-s + 2.35e4·14-s + (7.40e3 + 1.28e4i)16-s + (−1.06e4 + 1.84e4i)17-s + (−3.05e4 − 1.76e4i)19-s + (−414. − 239. i)20-s + (−4.05e4 + 7.02e4i)22-s + (5.08e3 + 8.80e3i)23-s + ⋯ |
| L(s) = 1 | + (1.25 − 0.722i)2-s + (0.544 − 0.942i)4-s − 0.0122i·5-s + (1.37 + 0.794i)7-s − 0.127i·8-s + (−0.00888 − 0.0153i)10-s + (−0.973 + 0.562i)11-s + (0.355 + 0.934i)13-s + 2.29·14-s + (0.452 + 0.783i)16-s + (−0.525 + 0.909i)17-s + (−1.02 − 0.589i)19-s + (−0.0115 − 0.00668i)20-s + (−0.812 + 1.40i)22-s + (0.0871 + 0.150i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.243917084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.243917084\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (-2.81e3 - 7.40e3i)T \) |
| good | 2 | \( 1 + (-14.1 + 8.17i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 + 3.43iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (-1.24e3 - 720. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (4.29e3 - 2.48e3i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.06e4 - 1.84e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (3.05e4 + 1.76e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-5.08e3 - 8.80e3i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-1.12e5 - 1.95e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 2.92e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (-2.33e5 + 1.34e5i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-6.38e5 + 3.68e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.27e5 - 2.20e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 5.34e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.24e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (2.55e6 + 1.47e6i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-4.29e5 + 7.44e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.64e5 + 9.47e4i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.61e6 - 9.33e5i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 4.31e4iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 1.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.89e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (1.63e6 - 9.44e5i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (3.59e6 + 2.07e6i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.44106782835038957326353833100, −11.15646907468780087615464830523, −10.87776852738556351189716335642, −8.980648954562239712005298438130, −7.964729409563374927362292604744, −6.21502959063058949662874922524, −4.95534398292163388908721087415, −4.32936578798611638502719806322, −2.55161248578624060315993140252, −1.74277693046066472316893609794,
0.831334353092614384205736816951, 2.88390667881517712468880731101, 4.39617754479175667989916388745, 5.10280713368794318474955557469, 6.31930173458133920685557593942, 7.59657783285346461190476209930, 8.361937668620316042621964035200, 10.34887232584644535554044563381, 11.13743804368751266256827401630, 12.47533479763275889704691141218
