| L(s) = 1 | + (0.694 + 3.93i)2-s + (0.677 − 3.84i)3-s + (−15.0 + 5.47i)4-s + (−48.1 − 40.3i)5-s + 15.6·6-s + (36.5 + 30.6i)7-s + (−32 − 55.4i)8-s + (214. + 77.9i)9-s + (125. − 217. i)10-s + (208. + 360. i)11-s + (10.8 + 61.5i)12-s + (830. − 302. i)13-s + (−95.3 + 165. i)14-s + (−187. + 157. i)15-s + (196. − 164. i)16-s + (819. + 298. i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.0434 − 0.246i)3-s + (−0.469 + 0.171i)4-s + (−0.860 − 0.722i)5-s + 0.177·6-s + (0.281 + 0.236i)7-s + (−0.176 − 0.306i)8-s + (0.880 + 0.320i)9-s + (0.397 − 0.688i)10-s + (0.518 + 0.898i)11-s + (0.0217 + 0.123i)12-s + (1.36 − 0.495i)13-s + (−0.130 + 0.225i)14-s + (−0.215 + 0.180i)15-s + (0.191 − 0.160i)16-s + (0.687 + 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.72704 + 0.577310i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72704 + 0.577310i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.694 - 3.93i)T \) |
| 37 | \( 1 + (3.52e3 - 7.54e3i)T \) |
| good | 3 | \( 1 + (-0.677 + 3.84i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (48.1 + 40.3i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-36.5 - 30.6i)T + (2.91e3 + 1.65e4i)T^{2} \) |
| 11 | \( 1 + (-208. - 360. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-830. + 302. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-819. - 298. i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 19 | \( 1 + (-95.0 + 539. i)T + (-2.32e6 - 8.46e5i)T^{2} \) |
| 23 | \( 1 + (64.9 - 112. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.25e3 - 3.90e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 3.67e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-4.23e3 + 1.54e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + 6.70e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-7.93e3 + 1.37e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-9.61e3 + 8.07e3i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (1.66e4 - 1.39e4i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (-2.71e4 + 9.87e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (4.11e4 + 3.45e4i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (1.17e4 - 6.67e4i)T + (-1.69e9 - 6.17e8i)T^{2} \) |
| 73 | \( 1 + 1.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-6.38e4 - 5.35e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-2.70e4 - 9.82e3i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (-1.78e4 + 1.50e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-2.56e4 + 4.43e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.65596059192275869358258511666, −12.67467163777340891962636001108, −11.84774923331096893715254963128, −10.22765014609609048758326688081, −8.713327485646157571215567621139, −7.900484948711195441386200411955, −6.71184342561015297082548944084, −5.04957757733646489333219499700, −3.90529064771691508530378008005, −1.18038890204583941708206155724,
1.09652144785041342096022352263, 3.37600434293443589289500659206, 4.20162831804341355205277483847, 6.24657233413174313363094574319, 7.72798847416199324713889290280, 9.052243345607480067820978795961, 10.41130869242677134925809173014, 11.25739164021689784842439540863, 12.08988413153575444711951330478, 13.53526343135150240746423334802
