| L(s) = 1 | + (135. − 234. i)5-s + (87.0 + 903. i)7-s + (2.75e3 + 4.76e3i)11-s + 8.80e3·13-s + (−3.42e3 − 5.92e3i)17-s + (1.32e4 − 2.29e4i)19-s + (−4.13e4 + 7.15e4i)23-s + (2.27e3 + 3.94e3i)25-s − 2.33e5·29-s + (1.98e4 + 3.43e4i)31-s + (2.23e5 + 1.02e5i)35-s + (−1.15e5 + 2.00e5i)37-s + 4.03e5·41-s − 1.22e5·43-s + (5.26e4 − 9.12e4i)47-s + ⋯ |
| L(s) = 1 | + (0.485 − 0.840i)5-s + (0.0958 + 0.995i)7-s + (0.623 + 1.08i)11-s + 1.11·13-s + (−0.168 − 0.292i)17-s + (0.442 − 0.766i)19-s + (−0.708 + 1.22i)23-s + (0.0291 + 0.0504i)25-s − 1.77·29-s + (0.119 + 0.207i)31-s + (0.883 + 0.402i)35-s + (−0.376 + 0.651i)37-s + 0.915·41-s − 0.235·43-s + (0.0740 − 0.128i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.283634217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.283634217\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-87.0 - 903. i)T \) |
| good | 5 | \( 1 + (-135. + 234. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-2.75e3 - 4.76e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 8.80e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (3.42e3 + 5.92e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.32e4 + 2.29e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (4.13e4 - 7.15e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 2.33e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.98e4 - 3.43e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.15e5 - 2.00e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 4.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-5.26e4 + 9.12e4i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (6.65e5 + 1.15e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-4.29e5 - 7.43e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-5.25e5 + 9.10e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.21e6 - 3.83e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.54e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.07e6 - 1.86e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (9.54e5 - 1.65e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 8.56e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (5.54e6 - 9.59e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.13e6T + 8.07e13T^{2} \) |
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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
−11.19640325581832939359958192502, −9.581755575902644256238848723170, −9.269524260548272717937236342178, −8.248063866070850962960865651319, −6.97116101111907788245056767700, −5.76563469634954154973555969194, −5.01910395470728396691356095894, −3.70786253110263471885046947897, −2.10753301742885692228728690664, −1.22954042517745053427139224459,
0.55113192819840708759356741865, 1.78588238399791664907258739073, 3.32570029981069953980962451997, 4.09723256190393111939473896514, 5.88010214425465464096629910883, 6.45368018395779362124162099255, 7.62628419746800561630221338643, 8.648551382687751260319997098238, 9.800882097447042648010204016446, 10.83010802990810267737714583264
