L(s) = 1 | + (2 − 2i)2-s + (−8.96 − 8.96i)3-s − 8i·4-s + (1.96 + 24.9i)5-s − 35.8·6-s + (53.1 − 53.1i)7-s + (−16 − 16i)8-s + 79.8i·9-s + (53.7 + 45.9i)10-s + 151.·11-s + (−71.7 + 71.7i)12-s + (162. + 162. i)13-s − 212. i·14-s + (205. − 241. i)15-s − 64·16-s + (130. − 130. i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (−0.996 − 0.996i)3-s − 0.5i·4-s + (0.0787 + 0.996i)5-s − 0.996·6-s + (1.08 − 1.08i)7-s + (−0.250 − 0.250i)8-s + 0.986i·9-s + (0.537 + 0.459i)10-s + 1.24·11-s + (−0.498 + 0.498i)12-s + (0.960 + 0.960i)13-s − 1.08i·14-s + (0.915 − 1.07i)15-s − 0.250·16-s + (0.450 − 0.450i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.213226842\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.213226842\) |
\(L(3)\) |
|
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 | \( 1 + (-1.96 - 24.9i)T \) |
| 23 | \( 1 + (-77.9 - 77.9i)T \) |
good | 3 | \( 1 + (8.96 + 8.96i)T + 81iT^{2} \) |
| 7 | \( 1 + (-53.1 + 53.1i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 - 151.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-162. - 162. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-130. + 130. i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 340. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 797. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.24e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (42.4 - 42.4i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.54e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (100. + 100. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (2.15e3 - 2.15e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (2.12e3 + 2.12e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + 6.38e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 826.T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-3.95e3 + 3.95e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 5.69e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-5.55e3 - 5.55e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 2.29e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (5.24e3 + 5.24e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 - 1.35e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-4.85e3 + 4.85e3i)T - 8.85e7iT^{2} \) |
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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
−11.57519590804114651815353309396, −10.84096246828144186346494304011, −9.590401118105800404862167784947, −7.901550641555010588895981195190, −6.68449847554845750988505509969, −6.46349461412844836436104575482, −4.85636703683496694499524821957, −3.68416203513899515906914788850, −1.79462983183435606662052734184, −0.863613895279208122097943165258,
1.34753573750148970789193515099, 3.74094116660397899920812933944, 4.79614247810040024613862518899, 5.51648407158568004649656012598, 6.18923871486477449935728222750, 8.177446359347695909886894669137, 8.726793252955034338279456830418, 9.956503751372785778219084618333, 11.13571122727875228753696225406, 11.94392390858964953226569557096