| L(s) = 1 | + (−3.06 + 2.57i)2-s + (17.5 + 6.39i)3-s + (2.77 − 15.7i)4-s + (12.8 + 73.1i)5-s + (−70.2 + 25.5i)6-s + (75.2 − 130. i)7-s + (32.0 + 55.4i)8-s + (81.4 + 68.3i)9-s + (−227. − 190. i)10-s + (388. + 673. i)11-s + (149. − 258. i)12-s + (−1.01e3 + 369. i)13-s + (104. + 592. i)14-s + (−240. + 1.36e3i)15-s + (−240. − 87.5i)16-s + (264. − 222. i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (1.12 + 0.410i)3-s + (0.0868 − 0.492i)4-s + (0.230 + 1.30i)5-s + (−0.796 + 0.289i)6-s + (0.580 − 1.00i)7-s + (0.176 + 0.306i)8-s + (0.335 + 0.281i)9-s + (−0.719 − 0.603i)10-s + (0.968 + 1.67i)11-s + (0.299 − 0.519i)12-s + (−1.66 + 0.607i)13-s + (0.142 + 0.808i)14-s + (−0.276 + 1.56i)15-s + (−0.234 − 0.0855i)16-s + (0.222 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0471 - 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0471 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.30416 + 1.24408i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30416 + 1.24408i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-1.54e3 - 322. i)T \) |
| good | 3 | \( 1 + (-17.5 - 6.39i)T + (186. + 156. i)T^{2} \) |
| 5 | \( 1 + (-12.8 - 73.1i)T + (-2.93e3 + 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-75.2 + 130. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-388. - 673. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (1.01e3 - 369. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-264. + 222. i)T + (2.46e5 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (-262. + 1.49e3i)T + (-6.04e6 - 2.20e6i)T^{2} \) |
| 29 | \( 1 + (2.85e3 + 2.39e3i)T + (3.56e6 + 2.01e7i)T^{2} \) |
| 31 | \( 1 + (-3.30e3 + 5.73e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (4.78e3 + 1.74e3i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 + (1.82e3 + 1.03e4i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-1.98e3 - 1.66e3i)T + (3.98e7 + 2.25e8i)T^{2} \) |
| 53 | \( 1 + (-4.01e3 + 2.27e4i)T + (-3.92e8 - 1.43e8i)T^{2} \) |
| 59 | \( 1 + (-3.98e3 + 3.34e3i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (2.65e3 - 1.50e4i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (-4.57e3 - 3.84e3i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-3.80e3 - 2.15e4i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 + (7.13e3 + 2.59e3i)T + (1.58e9 + 1.33e9i)T^{2} \) |
| 79 | \( 1 + (4.20e4 + 1.53e4i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (1.12e4 - 1.94e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-3.93e4 + 1.43e4i)T + (4.27e9 - 3.58e9i)T^{2} \) |
| 97 | \( 1 + (3.91e4 - 3.28e4i)T + (1.49e9 - 8.45e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.98471921606942957345247659691, −14.68220637377088557860512844657, −13.95799517878548645854793259050, −11.69314981143523743407908318586, −10.01261339117293103921433528754, −9.590736967223193672710598340367, −7.62517429577163397798785701153, −6.96370361045343975670494323300, −4.29480108155490786752336370201, −2.29561732479272181474521253171,
1.26793158345256148367139641055, 2.93361391906383609929797715567, 5.32505489124285382339132218253, 7.85445796921399626215352003333, 8.743725155036067643523796340935, 9.412874455361275014455824630404, 11.56466033931603467803878368637, 12.53787291615263305196057481121, 13.72076553918407351396170328931, 14.82062635982439547301492911603
