| L(s) = 1 | + (0.00202 − 0.0283i)2-s + (4.96 − 1.07i)3-s + (7.91 + 1.13i)4-s + (7.06 − 8.66i)5-s + (−0.0205 − 0.142i)6-s + (8.60 − 4.70i)7-s + (0.0964 − 0.443i)8-s + (−1.09 + 0.499i)9-s + (−0.231 − 0.217i)10-s + (−28.4 − 24.6i)11-s + (40.5 − 2.89i)12-s + (−28.4 + 52.1i)13-s + (−0.115 − 0.253i)14-s + (25.6 − 50.6i)15-s + (61.3 + 18.0i)16-s + (−52.8 − 39.5i)17-s + ⋯ |
| L(s) = 1 | + (0.000715 − 0.0100i)2-s + (0.955 − 0.207i)3-s + (0.989 + 0.142i)4-s + (0.631 − 0.775i)5-s + (−0.00139 − 0.00970i)6-s + (0.464 − 0.253i)7-s + (0.00426 − 0.0196i)8-s + (−0.0405 + 0.0184i)9-s + (−0.00730 − 0.00687i)10-s + (−0.778 − 0.674i)11-s + (0.974 − 0.0697i)12-s + (−0.607 + 1.11i)13-s + (−0.00220 − 0.00483i)14-s + (0.442 − 0.871i)15-s + (0.959 + 0.281i)16-s + (−0.753 − 0.564i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.463i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.885 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.65279 - 0.652302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65279 - 0.652302i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-7.06 + 8.66i)T \) |
| 23 | \( 1 + (-95.7 - 54.8i)T \) |
| good | 2 | \( 1 + (-0.00202 + 0.0283i)T + (-7.91 - 1.13i)T^{2} \) |
| 3 | \( 1 + (-4.96 + 1.07i)T + (24.5 - 11.2i)T^{2} \) |
| 7 | \( 1 + (-8.60 + 4.70i)T + (185. - 288. i)T^{2} \) |
| 11 | \( 1 + (28.4 + 24.6i)T + (189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (28.4 - 52.1i)T + (-1.18e3 - 1.84e3i)T^{2} \) |
| 17 | \( 1 + (52.8 + 39.5i)T + (1.38e3 + 4.71e3i)T^{2} \) |
| 19 | \( 1 + (10.2 - 71.6i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-205. + 29.5i)T + (2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-69.8 + 44.8i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (84.1 + 31.3i)T + (3.82e4 + 3.31e4i)T^{2} \) |
| 41 | \( 1 + (187. - 411. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-28.4 - 130. i)T + (-7.23e4 + 3.30e4i)T^{2} \) |
| 47 | \( 1 + (-2.82 - 2.82i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (212. + 389. i)T + (-8.04e4 + 1.25e5i)T^{2} \) |
| 59 | \( 1 + (110. + 376. i)T + (-1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-72.3 - 112. i)T + (-9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (774. + 55.4i)T + (2.97e5 + 4.28e4i)T^{2} \) |
| 71 | \( 1 + (51.6 + 59.6i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (200. + 267. i)T + (-1.09e5 + 3.73e5i)T^{2} \) |
| 79 | \( 1 + (-1.16e3 + 341. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-68.0 + 182. i)T + (-4.32e5 - 3.74e5i)T^{2} \) |
| 89 | \( 1 + (-1.08e3 - 699. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (393. + 1.05e3i)T + (-6.89e5 + 5.97e5i)T^{2} \) |
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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
−13.22621706078917599992023317541, −11.98145897012175609367812678219, −11.00561045401101704911376020949, −9.690127453593979176193492202696, −8.532567926720740788638337475983, −7.73367616126291104005338171560, −6.38036103475795891662682131252, −4.87820232848311837738260683030, −2.88753912043129502739505572200, −1.73451865934905575453566403270,
2.27573761067282386370667001345, 2.94693095924531259066814535432, 5.19708830145785682161004585580, 6.61391250689799973166856220078, 7.66697054545164541167563841450, 8.832434953598528432569141348783, 10.24606758314848385705346299580, 10.74684188289416711459528073471, 12.13051741210286235318499012087, 13.32912825219443086538257077622
