| L(s) = 1 | + (1.39 − 6.10i)2-s + (14.8 − 18.6i)3-s + (−6.45 − 3.10i)4-s + (−16.2 + 20.3i)5-s + (−92.8 − 116. i)6-s + (7.62 − 129. i)7-s + (96.9 − 121. i)8-s + (−71.9 − 315. i)9-s + (101. + 127. i)10-s + (−78.7 + 345. i)11-s + (−153. + 73.9i)12-s + (−113. + 498. i)13-s + (−778. − 226. i)14-s + (137. + 604. i)15-s + (−749. − 939. i)16-s + (364. − 175. i)17-s + ⋯ |
| L(s) = 1 | + (0.246 − 1.07i)2-s + (0.951 − 1.19i)3-s + (−0.201 − 0.0971i)4-s + (−0.290 + 0.364i)5-s + (−1.05 − 1.32i)6-s + (0.0588 − 0.998i)7-s + (0.535 − 0.671i)8-s + (−0.295 − 1.29i)9-s + (0.321 + 0.403i)10-s + (−0.196 + 0.859i)11-s + (−0.307 + 0.148i)12-s + (−0.186 + 0.818i)13-s + (−1.06 − 0.309i)14-s + (0.158 + 0.693i)15-s + (−0.731 − 0.917i)16-s + (0.305 − 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.744785 - 2.46209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744785 - 2.46209i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-7.62 + 129. i)T \) |
| good | 2 | \( 1 + (-1.39 + 6.10i)T + (-28.8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (-14.8 + 18.6i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (16.2 - 20.3i)T + (-695. - 3.04e3i)T^{2} \) |
| 11 | \( 1 + (78.7 - 345. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (113. - 498. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-364. + 175. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 - 706.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-205. - 99.0i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 29 | \( 1 + (367. - 176. i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 + 1.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.22e4 + 5.92e3i)T + (4.32e7 - 5.42e7i)T^{2} \) |
| 41 | \( 1 + (-1.21e4 + 1.52e4i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (7.73e3 + 9.70e3i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (5.53e3 - 2.42e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-1.62e4 - 7.80e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-2.34e4 - 2.94e4i)T + (-1.59e8 + 6.96e8i)T^{2} \) |
| 61 | \( 1 + (867. - 417. i)T + (5.26e8 - 6.60e8i)T^{2} \) |
| 67 | \( 1 + 6.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (3.62e4 + 1.74e4i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-5.57e3 - 2.44e4i)T + (-1.86e9 + 8.99e8i)T^{2} \) |
| 79 | \( 1 + 7.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-1.87e4 - 8.23e4i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-2.22e4 - 9.76e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 + 1.08e5T + 8.58e9T^{2} \) |
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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
−13.78283520177512219291239879350, −12.95201920210721415118032224239, −11.99334761966012781423010221044, −10.80861106240912175449080647525, −9.445438499534733180900392418070, −7.51128806053543616640126209549, −7.11320947681783861521037717691, −4.03573050255573669180735315181, −2.60988287142702634580566631045, −1.29334670718605639392402474017,
2.92999172978303428629747707157, 4.72873227441195447013340579651, 5.88826903378507940110600581316, 7.985669491641776528096058087712, 8.640703953733499327955947244687, 10.01536365535620961920856997593, 11.45883946276008173598920994324, 13.21551429501209068242812197308, 14.59213581657705610464068989673, 15.02770375898081186176587022292
