| L(s) = 1 | + (−3.75 − 1.36i)2-s + (7.77 − 2.83i)3-s + (12.2 + 10.2i)4-s + (−14.8 + 83.9i)5-s − 33.1·6-s + (8.29 − 47.0i)7-s + (−32.0 − 55.4i)8-s + (−133. + 112. i)9-s + (170. − 295. i)10-s + (−211. − 366. i)11-s + (124. + 45.3i)12-s + (−333. − 280. i)13-s + (−95.5 + 165. i)14-s + (122. + 695. i)15-s + (44.4 + 252. i)16-s + (1.16e3 − 973. i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.499 − 0.181i)3-s + (0.383 + 0.321i)4-s + (−0.264 + 1.50i)5-s − 0.375·6-s + (0.0639 − 0.362i)7-s + (−0.176 − 0.306i)8-s + (−0.549 + 0.461i)9-s + (0.539 − 0.934i)10-s + (−0.527 − 0.914i)11-s + (0.249 + 0.0908i)12-s + (−0.548 − 0.459i)13-s + (−0.130 + 0.225i)14-s + (0.140 + 0.797i)15-s + (0.0434 + 0.246i)16-s + (0.974 − 0.817i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0232591 + 0.226129i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0232591 + 0.226129i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.75 + 1.36i)T \) |
| 37 | \( 1 + (-4.76e3 - 6.82e3i)T \) |
| good | 3 | \( 1 + (-7.77 + 2.83i)T + (186. - 156. i)T^{2} \) |
| 5 | \( 1 + (14.8 - 83.9i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-8.29 + 47.0i)T + (-1.57e4 - 5.74e3i)T^{2} \) |
| 11 | \( 1 + (211. + 366. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (333. + 280. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-1.16e3 + 973. i)T + (2.46e5 - 1.39e6i)T^{2} \) |
| 19 | \( 1 + (2.78e3 - 1.01e3i)T + (1.89e6 - 1.59e6i)T^{2} \) |
| 23 | \( 1 + (1.29e3 - 2.25e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (3.30e3 + 5.71e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 7.57e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-1.43e4 - 1.20e4i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + 907.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.60e3 - 1.14e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (5.34e3 + 3.02e4i)T + (-3.92e8 + 1.43e8i)T^{2} \) |
| 59 | \( 1 + (4.33e3 + 2.45e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (1.52e4 + 1.27e4i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (4.18e3 - 2.37e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (2.77e4 - 1.00e4i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 + 1.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.30e4 - 7.41e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-5.20e4 + 4.36e4i)T + (6.84e8 - 3.87e9i)T^{2} \) |
| 89 | \( 1 + (-3.97e3 - 2.25e4i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (6.04e3 - 1.04e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−14.28459480901099840543489014348, −13.09404706384397022520072797099, −11.43866440290011260966294934407, −10.79685239083223007300902593280, −9.756043297616441041952185836150, −8.051376962344916695174378052379, −7.54554081670055380470779690614, −5.98021540934677005768008856820, −3.42970336667661466008169627765, −2.40096459451617922147649990601,
0.10662806447351446907156714472, 2.05006449248182130317971135040, 4.31317223148053846657899597798, 5.72574733030614964871832788752, 7.54895791145550748729330954603, 8.748034659669921100994400339299, 9.149245713105219280558621237252, 10.59238787082465239411926925781, 12.24863524337627931070467419219, 12.73427559175505057962706365235
