L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s − 0.999·6-s + (1.15 − 0.837i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (4.35 − 3.16i)11-s + (−0.309 − 0.951i)12-s + (2.13 − 6.56i)13-s + (1.15 + 0.837i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (0.246 + 0.179i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.178 + 0.549i)3-s + (−0.404 + 0.293i)4-s + 0.447·5-s − 0.408·6-s + (0.435 − 0.316i)7-s + (−0.286 − 0.207i)8-s + (−0.269 − 0.195i)9-s + (0.0977 + 0.300i)10-s + (1.31 − 0.953i)11-s + (−0.0892 − 0.274i)12-s + (0.591 − 1.82i)13-s + (0.308 + 0.223i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.0598 + 0.0434i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82752 + 0.393374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82752 + 0.393374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + (3.97 - 3.89i)T \) |
good | 7 | \( 1 + (-1.15 + 0.837i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-4.35 + 3.16i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.13 + 6.56i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.246 - 0.179i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.20 + 3.72i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.909 + 0.661i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.52 + 4.70i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 + (-0.985 - 3.03i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.344 - 1.05i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (3.46 - 10.6i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.509 + 0.369i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.93 + 12.1i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 - 1.85T + 61T^{2} \) |
| 67 | \( 1 + 6.25T + 67T^{2} \) |
| 71 | \( 1 + (-11.5 - 8.40i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (13.0 - 9.45i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (12.7 + 9.28i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.31 - 13.2i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-10.6 + 7.73i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.07 + 3.69i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.05741303201146580105593026403, −9.174264822882313654291336679112, −8.447421741603059238726152265247, −7.66222685096468835732544527150, −6.39087898509270489327821196345, −5.89878438220730141351403753928, −4.95940168639589030427840299496, −3.94743431622977189061495129714, −3.02191993563212586335758271225, −0.944985133563604803090513383712,
1.61298095472237521362769035246, 1.97550393562791628741911943565, 3.74584826357178766240441445600, 4.52107907244601068525278703509, 5.70357535120753290635538751672, 6.52837189146723365800859728365, 7.30303566476071964589850077761, 8.663123787808629849024554091689, 9.185738540749537990109442581668, 9.999181881263967004866723416314