L(s) = 1 | + (−3.08 + 4.73i)2-s + (14.4 + 5.82i)3-s + (−12.9 − 29.2i)4-s + (−43.0 − 35.7i)5-s + (−72.2 + 50.5i)6-s + 168.·7-s + (178. + 29.2i)8-s + (175. + 168. i)9-s + (302. − 93.5i)10-s + 165.·11-s + (−16.1 − 498. i)12-s − 182. i·13-s + (−519. + 796. i)14-s + (−413. − 766. i)15-s + (−690. + 756. i)16-s + 1.80e3·17-s + ⋯ |
L(s) = 1 | + (−0.546 + 0.837i)2-s + (0.927 + 0.373i)3-s + (−0.403 − 0.914i)4-s + (−0.769 − 0.638i)5-s + (−0.819 + 0.572i)6-s + 1.29·7-s + (0.986 + 0.161i)8-s + (0.720 + 0.693i)9-s + (0.955 − 0.295i)10-s + 0.412·11-s + (−0.0323 − 0.999i)12-s − 0.299i·13-s + (−0.708 + 1.08i)14-s + (−0.474 − 0.879i)15-s + (−0.674 + 0.738i)16-s + 1.51·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.61819 + 0.791946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61819 + 0.791946i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.08 - 4.73i)T \) |
| 3 | \( 1 + (-14.4 - 5.82i)T \) |
| 5 | \( 1 + (43.0 + 35.7i)T \) |
good | 7 | \( 1 - 168.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 165.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 182. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.17e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 391. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 382. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 8.73e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.38e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.35e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.60e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 8.22e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.47e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.02e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.46e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 2.54e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.57e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5iT - 8.58e9T^{2} \) |
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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
−14.66489895493584255647463522568, −13.59487338130404052133678815786, −11.92788945399281522219765049446, −10.46023412965044324148777231589, −9.197820349909629695317078110540, −8.087195569698329241813866731534, −7.65621311709263546671216227717, −5.32873843062797775541054506062, −4.08225955484749724006231799352, −1.34139304396172004203477791379,
1.30543451497192449131845114278, 2.93061737473161203166477674589, 4.29472505037322256846534280967, 7.26224284481163068008743065680, 8.025182353761326640284939059242, 9.099709516426401639854624133242, 10.56290294586080782135411279744, 11.62939930519537034772889442057, 12.50787652803299814700319010902, 14.11269278344166479877709529712