L(s) = 1 | + (2 + 2i)2-s + (−5.57 + 5.57i)3-s + 8i·4-s + (−24.4 − 5.04i)5-s − 22.3·6-s + (−57.8 − 57.8i)7-s + (−16 + 16i)8-s + 18.7i·9-s + (−38.8 − 59.0i)10-s − 17.6·11-s + (−44.6 − 44.6i)12-s + (199. − 199. i)13-s − 231. i·14-s + (164. − 108. i)15-s − 64·16-s + (249. + 249. i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (−0.619 + 0.619i)3-s + 0.5i·4-s + (−0.979 − 0.201i)5-s − 0.619·6-s + (−1.18 − 1.18i)7-s + (−0.250 + 0.250i)8-s + 0.231i·9-s + (−0.388 − 0.590i)10-s − 0.145·11-s + (−0.309 − 0.309i)12-s + (1.18 − 1.18i)13-s − 1.18i·14-s + (0.732 − 0.482i)15-s − 0.250·16-s + (0.862 + 0.862i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.183036778\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.183036778\) |
\(L(3)\) |
|
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 | \( 1 + (24.4 + 5.04i)T \) |
| 23 | \( 1 + (77.9 - 77.9i)T \) |
good | 3 | \( 1 + (5.57 - 5.57i)T - 81iT^{2} \) |
| 7 | \( 1 + (57.8 + 57.8i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 17.6T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-199. + 199. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-249. - 249. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 511. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 1.03e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 41.1T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-234. - 234. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.36e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.85e3 + 1.85e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-185. - 185. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.76e3 + 1.76e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + 4.60e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.86e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-6.12e3 - 6.12e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 1.52e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-4.60e3 + 4.60e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 8.47e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (3.74e3 - 3.74e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.43e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-4.19e3 - 4.19e3i)T + 8.85e7iT^{2} \) |
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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
−11.61562127690505351752840252032, −10.52062423746777988663609582262, −10.06697139956521843602165965122, −8.249813487024335845864545691221, −7.64375449194747393779493271046, −6.29304519596883826025026701671, −5.41910409551353594855429871717, −3.88163273072309897580630356974, −3.63301338186520162445198043749, −0.54974800944292500864198597496,
0.834702673133287096274802206307, 2.73462357461777085864640376253, 3.76535551329488859843354137488, 5.30842508093877352315442970184, 6.42951054589258735472035383958, 7.02339368978940438639438291039, 8.757634543373014025596081440793, 9.499366930747635440795040984131, 11.06195316550717377244269275526, 11.64509378856257946097222762227