L(s) = 1 | + (−0.694 + 3.93i)2-s + (15.2 + 12.7i)3-s + (−15.0 − 5.47i)4-s + (−77.4 + 28.2i)5-s + (−60.8 + 51.0i)6-s + (31.0 + 53.8i)7-s + (32 − 55.4i)8-s + (26.2 + 148. i)9-s + (−57.2 − 324. i)10-s + (23.0 − 39.8i)11-s + (−158. − 275. i)12-s + (−470. + 394. i)13-s + (−233. + 85.0i)14-s + (−1.53e3 − 560. i)15-s + (196. + 164. i)16-s + (−265. + 1.50e3i)17-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.975 + 0.818i)3-s + (−0.469 − 0.171i)4-s + (−1.38 + 0.504i)5-s + (−0.689 + 0.578i)6-s + (0.239 + 0.415i)7-s + (0.176 − 0.306i)8-s + (0.108 + 0.613i)9-s + (−0.181 − 1.02i)10-s + (0.0573 − 0.0994i)11-s + (−0.318 − 0.551i)12-s + (−0.772 + 0.648i)13-s + (−0.318 + 0.116i)14-s + (−1.76 − 0.642i)15-s + (0.191 + 0.160i)16-s + (−0.223 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.149816 + 1.29404i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.149816 + 1.29404i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.694 - 3.93i)T \) |
| 19 | \( 1 + (-1.05e3 + 1.16e3i)T \) |
good | 3 | \( 1 + (-15.2 - 12.7i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (77.4 - 28.2i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-31.0 - 53.8i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-23.0 + 39.8i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (470. - 394. i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (265. - 1.50e3i)T + (-1.33e6 - 4.85e5i)T^{2} \) |
| 23 | \( 1 + (-1.54e3 - 563. i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-1.41e3 - 8.00e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-3.13e3 - 5.43e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 7.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (3.98e3 + 3.33e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-7.85e3 + 2.86e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (1.31e3 + 7.47e3i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 + (2.46e4 + 8.97e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (8.90e3 - 5.05e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (1.84e4 + 6.71e3i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (1.06e4 + 6.05e4i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-4.44e4 + 1.61e4i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 + (3.55e4 + 2.97e4i)T + (3.59e8 + 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-4.19e4 - 3.51e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-2.24e4 - 3.89e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (8.75e4 - 7.34e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-2.28e4 + 1.29e5i)T + (-8.06e9 - 2.93e9i)T^{2} \) |
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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
−15.42524563571942205281734604938, −15.03437099855187636861968319019, −14.08672538341959009946245453582, −12.20685509335219147607276157877, −10.76738951876112815812140511670, −9.201527665232353319003801984308, −8.271836521320867514241746618511, −7.00706382528802995347471806328, −4.64389830317515501106124250616, −3.29608478183892301773904021913,
0.72630314689393747987219716036, 2.82356326434407814653636847141, 4.48678265702470795247436461461, 7.57762259149629517468690294783, 8.036627994977716414191390772967, 9.556977288109668247900490507302, 11.37290737888065568641157285292, 12.32039858146367615352999424055, 13.37312035470890899222218090463, 14.49197088697577311419981800679