L(s) = 1 | + (−2 − 2i)2-s + (−6.81 + 6.81i)3-s + 8i·4-s + (24.3 − 5.62i)5-s + 27.2·6-s + (−23.8 − 23.8i)7-s + (16 − 16i)8-s − 11.9i·9-s + (−59.9 − 37.4i)10-s − 79.3·11-s + (−54.5 − 54.5i)12-s + (−18.8 + 18.8i)13-s + 95.3i·14-s + (−127. + 204. i)15-s − 64·16-s + (178. + 178. i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (−0.757 + 0.757i)3-s + 0.5i·4-s + (0.974 − 0.224i)5-s + 0.757·6-s + (−0.486 − 0.486i)7-s + (0.250 − 0.250i)8-s − 0.147i·9-s + (−0.599 − 0.374i)10-s − 0.655·11-s + (−0.378 − 0.378i)12-s + (−0.111 + 0.111i)13-s + 0.486i·14-s + (−0.567 + 0.908i)15-s − 0.250·16-s + (0.616 + 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3775596990\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3775596990\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 + 2i)T \) |
| 5 | \( 1 + (-24.3 + 5.62i)T \) |
| 23 | \( 1 + (-77.9 + 77.9i)T \) |
good | 3 | \( 1 + (6.81 - 6.81i)T - 81iT^{2} \) |
| 7 | \( 1 + (23.8 + 23.8i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 79.3T + 1.46e4T^{2} \) |
| 13 | \( 1 + (18.8 - 18.8i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-178. - 178. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 53.2iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 491. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 739.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (830. + 830. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.96e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.35e3 + 1.35e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (2.26e3 + 2.26e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.90e3 + 1.90e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + 1.01e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 3.14e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (1.46e3 + 1.46e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 6.52e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-2.15e3 + 2.15e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 6.66e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-1.68e3 + 1.68e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 3.80e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (548. + 548. i)T + 8.85e7iT^{2} \) |
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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
−10.71337043806152455693338809548, −10.35869134083153763652103196136, −9.614236079407990520751517035359, −8.507952022362070050585405987727, −7.12822397993732499942195920909, −5.81592344856039217033087437412, −4.91489115400952077156270001895, −3.49922591405965164564809033192, −1.87253168323410813994483455395, −0.17092465953803871280801879327,
1.30872313057049731425427201295, 2.79837071770145170071732399376, 5.21466140883037171829240936487, 5.94156915118193790410685779409, 6.75086226858562696030534554109, 7.69919948948026970529281735642, 9.097090460153469373290724095697, 9.848173071345426030871973346116, 10.84816473581377507851866042863, 11.94333143204311911605410986693