| L(s) = 1 | + (−5.18 − 0.406i)3-s + (−8.68 + 15.0i)5-s + (6.03 + 17.5i)7-s + (26.6 + 4.21i)9-s + (−9.11 + 5.26i)11-s + (−11.0 + 6.35i)13-s + (51.1 − 74.4i)15-s + (−28.9 + 50.2i)17-s + (56.6 − 32.7i)19-s + (−24.1 − 93.1i)21-s + (−31.2 − 18.0i)23-s + (−88.3 − 153. i)25-s + (−136. − 32.6i)27-s + (−230. − 133. i)29-s + 162. i·31-s + ⋯ |
| L(s) = 1 | + (−0.996 − 0.0782i)3-s + (−0.776 + 1.34i)5-s + (0.326 + 0.945i)7-s + (0.987 + 0.156i)9-s + (−0.249 + 0.144i)11-s + (−0.235 + 0.135i)13-s + (0.879 − 1.28i)15-s + (−0.413 + 0.716i)17-s + (0.684 − 0.395i)19-s + (−0.251 − 0.967i)21-s + (−0.283 − 0.163i)23-s + (−0.707 − 1.22i)25-s + (−0.972 − 0.232i)27-s + (−1.47 − 0.852i)29-s + 0.942i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.606i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2754383910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2754383910\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.18 + 0.406i)T \) |
| 7 | \( 1 + (-6.03 - 17.5i)T \) |
| good | 5 | \( 1 + (8.68 - 15.0i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (9.11 - 5.26i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (11.0 - 6.35i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (28.9 - 50.2i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-56.6 + 32.7i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (31.2 + 18.0i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (230. + 133. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 162. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (59.1 + 102. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-33.7 - 58.5i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-136. + 235. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (39.3 + 22.7i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 - 427.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 896. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 72.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 664. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (363. + 209. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (711. - 1.23e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-759. - 1.31e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.01e3 + 584. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.93835371615553173324210419861, −11.23051417869378315672531830835, −10.63935150317572492639524857869, −9.478787423011018785680182154864, −8.003621945524256724952981665117, −7.12020547160353696973975141746, −6.20351039642835367594806523849, −5.10801765507481778653268829313, −3.72270526919838285725668693234, −2.18592181713305423608381682013,
0.13816179796873030437285153524, 1.23190164244639347809400102129, 3.88959175934149031789698253545, 4.75075260258642669431304727587, 5.60187969207912097112811336570, 7.19206528507893763302972627093, 7.86705395569443963894846967842, 9.155860841417701794375959728650, 10.16742450993575190531623364853, 11.29250563021190425166579584762
