| L(s) = 1 | + (−1.41 + 2.44i)2-s + (4.5 − 2.59i)3-s + (−3.99 − 6.92i)4-s + (12.2 + 7.07i)5-s + 14.6i·6-s + 22.6·8-s + (13.5 − 23.3i)9-s + (−34.6 + 20.0i)10-s + (−32.0 − 55.4i)11-s + (−35.9 − 20.7i)12-s + 228. i·13-s + 73.5·15-s + (−32.0 + 55.4i)16-s + (195. − 112. i)17-s + (38.1 + 66.1i)18-s + (255. + 147. i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.490 + 0.283i)5-s + 0.408i·6-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.346 + 0.200i)10-s + (−0.264 − 0.458i)11-s + (−0.249 − 0.144i)12-s + 1.35i·13-s + 0.326·15-s + (−0.125 + 0.216i)16-s + (0.675 − 0.390i)17-s + (0.117 + 0.204i)18-s + (0.707 + 0.408i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.067180218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067180218\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 2.44i)T \) |
| 3 | \( 1 + (-4.5 + 2.59i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-12.2 - 7.07i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (32.0 + 55.4i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 228. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (-195. + 112. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-255. - 147. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (354. - 614. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 740.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (-577. + 333. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-416. + 722. i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 2.81e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.06e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (-531. - 306. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-576. - 998. i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-3.02e3 + 1.74e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.96e3 + 1.13e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-4.33e3 - 7.51e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 353.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (3.52e3 - 2.03e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (3.23e3 - 5.60e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 8.22e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-1.34e4 - 7.76e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + 1.55e3iT - 8.85e7T^{2} \) |
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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
−11.28212403619325587640923991338, −9.915673757106641555727900193994, −9.475694541360336508490032715562, −8.301194220333191498261096484692, −7.50332029936849949017100700783, −6.47353393516312535788808696758, −5.58916623243219036350298363910, −4.09485916710807428931128432946, −2.57274615779872627048929628806, −1.17604427483482467694470579793,
0.800005105726545056673302776021, 2.28132592752551774343121354601, 3.34213038147860794319822553799, 4.69940988990137196538915859048, 5.80060716221685761034717794952, 7.43098842109967181086921581886, 8.269790156551313824062818659319, 9.191607003737086387356342519927, 10.18045358696225477805850846746, 10.53663191740667678995905537708
