| L(s) = 1 | + (−0.967 + 2.65i)2-s + (1.72 − 0.304i)3-s + (−6.12 − 5.14i)4-s + (−25.8 + 21.7i)5-s + (−0.861 + 4.88i)6-s + (−9.12 + 15.8i)7-s + (19.5 − 11.3i)8-s + (−73.2 + 26.6i)9-s + (−32.6 − 89.8i)10-s + (4.73 + 8.19i)11-s + (−12.1 − 7.01i)12-s + (−190. − 33.6i)13-s + (−33.1 − 39.5i)14-s + (−38.1 + 45.4i)15-s + (11.1 + 63.0i)16-s + (407. + 148. i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (0.191 − 0.0338i)3-s + (−0.383 − 0.321i)4-s + (−1.03 + 0.868i)5-s + (−0.0239 + 0.135i)6-s + (−0.186 + 0.322i)7-s + (0.306 − 0.176i)8-s + (−0.903 + 0.329i)9-s + (−0.326 − 0.898i)10-s + (0.0390 + 0.0677i)11-s + (−0.0844 − 0.0487i)12-s + (−1.12 − 0.199i)13-s + (−0.169 − 0.201i)14-s + (−0.169 + 0.201i)15-s + (0.0434 + 0.246i)16-s + (1.41 + 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0671223 + 0.649629i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0671223 + 0.649629i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.967 - 2.65i)T \) |
| 19 | \( 1 + (-296. - 206. i)T \) |
| good | 3 | \( 1 + (-1.72 + 0.304i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (25.8 - 21.7i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (9.12 - 15.8i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-4.73 - 8.19i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (190. + 33.6i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-407. - 148. i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-185. - 155. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-276. - 760. i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (648. + 374. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.72e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-159. + 28.0i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (2.36e3 - 1.98e3i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (1.58e3 - 577. i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-767. + 914. i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-863. + 2.37e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (3.07e3 + 2.57e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-1.34e3 - 3.70e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-4.92e3 - 5.87e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-364. - 2.06e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (9.22e3 - 1.62e3i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (4.59e3 - 7.95e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-9.50e3 - 1.67e3i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (1.74e3 - 4.78e3i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−16.04346757446906869686287947011, −14.76781789751912422341924659506, −14.38110240729320848126251247067, −12.45026880600155090488302478257, −11.21205032924939242108464373898, −9.770326816630532066341904623889, −8.127921738300994887861866677660, −7.26556538541194499532509815384, −5.49990077722338736294588848513, −3.25947933124648653940938297767,
0.46966952042797254353031264905, 3.26607592637358483705359781551, 4.95295230140911546087605432875, 7.51825121952745543380274030613, 8.707114320588784694255881509999, 9.913512008403914970331477500991, 11.69426028333847207791336689403, 12.15621495377125200801424337593, 13.67046607697937032864783738409, 14.96392726046338815085613058468
