| L(s) = 1 | + (3.06 − 2.57i)2-s + (18.6 + 6.79i)3-s + (2.77 − 15.7i)4-s + (18.3 + 104. i)5-s + (74.6 − 27.1i)6-s + (−18.2 + 31.5i)7-s + (−32.0 − 55.4i)8-s + (116. + 97.6i)9-s + (323. + 271. i)10-s + (−257. − 445. i)11-s + (158. − 275. i)12-s + (295. − 107. i)13-s + (25.3 + 143. i)14-s + (−364. + 2.06e3i)15-s + (−240. − 87.5i)16-s + (1.75e3 − 1.46e3i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (1.19 + 0.436i)3-s + (0.0868 − 0.492i)4-s + (0.328 + 1.86i)5-s + (0.847 − 0.308i)6-s + (−0.140 + 0.243i)7-s + (−0.176 − 0.306i)8-s + (0.478 + 0.401i)9-s + (1.02 + 0.858i)10-s + (−0.641 − 1.11i)11-s + (0.318 − 0.552i)12-s + (0.484 − 0.176i)13-s + (0.0345 + 0.195i)14-s + (−0.418 + 2.37i)15-s + (−0.234 − 0.0855i)16-s + (1.46 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.90256 + 0.468583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.90256 + 0.468583i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.06 + 2.57i)T \) |
| 19 | \( 1 + (251. + 1.55e3i)T \) |
| good | 3 | \( 1 + (-18.6 - 6.79i)T + (186. + 156. i)T^{2} \) |
| 5 | \( 1 + (-18.3 - 104. i)T + (-2.93e3 + 1.06e3i)T^{2} \) |
| 7 | \( 1 + (18.2 - 31.5i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (257. + 445. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-295. + 107. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-1.75e3 + 1.46e3i)T + (2.46e5 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (350. - 1.98e3i)T + (-6.04e6 - 2.20e6i)T^{2} \) |
| 29 | \( 1 + (854. + 716. i)T + (3.56e6 + 2.01e7i)T^{2} \) |
| 31 | \( 1 + (2.58e3 - 4.48e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 4.29e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.99e3 - 727. i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-1.17e3 - 6.68e3i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (2.00e3 + 1.68e3i)T + (3.98e7 + 2.25e8i)T^{2} \) |
| 53 | \( 1 + (-358. + 2.03e3i)T + (-3.92e8 - 1.43e8i)T^{2} \) |
| 59 | \( 1 + (2.35e4 - 1.97e4i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (4.24e3 - 2.40e4i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (1.54e4 + 1.29e4i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (1.04e4 + 5.92e4i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 + (-3.84e4 - 1.39e4i)T + (1.58e9 + 1.33e9i)T^{2} \) |
| 79 | \( 1 + (7.76e4 + 2.82e4i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-2.15e4 + 3.73e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (8.94e4 - 3.25e4i)T + (4.27e9 - 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-7.95e4 + 6.67e4i)T + (1.49e9 - 8.45e9i)T^{2} \) |
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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
−14.99506317933348673775630785495, −14.09090912236673816821741944056, −13.54951708932033355173226012260, −11.46657814312729446381684712130, −10.45834213281799394892789543816, −9.334157974330127948857786602284, −7.55762358362474914063756661562, −5.86325565803428149864511654131, −3.28268761568012469606597720376, −2.80821262071275489450421084088,
1.75233951222987119927881956637, 4.04643017784869885489238252047, 5.62204230995754193136587840848, 7.77810957408623513056194867184, 8.491476061855665533095186530620, 9.825595173017000038439313211813, 12.51523228989596649009556984218, 12.82623331118911557675415638953, 13.95141596303263529796476682706, 15.01904645407214642715487214092
