| L(s) = 1 | + (−0.530 + 0.919i)2-s + (2.96 − 0.483i)3-s + (1.43 + 2.48i)4-s + (−3.80 + 3.23i)5-s + (−1.12 + 2.97i)6-s + (−3.04 + 6.30i)7-s − 7.29·8-s + (8.53 − 2.86i)9-s + (−0.956 − 5.22i)10-s + (9.99 − 5.76i)11-s + (5.45 + 6.67i)12-s − 5.56i·13-s + (−4.17 − 6.14i)14-s + (−9.70 + 11.4i)15-s + (−1.87 + 3.24i)16-s + (7.82 + 13.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.265 + 0.459i)2-s + (0.986 − 0.161i)3-s + (0.359 + 0.622i)4-s + (−0.761 + 0.647i)5-s + (−0.187 + 0.496i)6-s + (−0.435 + 0.900i)7-s − 0.912·8-s + (0.948 − 0.318i)9-s + (−0.0956 − 0.522i)10-s + (0.908 − 0.524i)11-s + (0.454 + 0.555i)12-s − 0.428i·13-s + (−0.298 − 0.439i)14-s + (−0.647 + 0.762i)15-s + (−0.117 + 0.202i)16-s + (0.460 + 0.797i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0438 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0438 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.08241 + 1.03592i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08241 + 1.03592i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.96 + 0.483i)T \) |
| 5 | \( 1 + (3.80 - 3.23i)T \) |
| 7 | \( 1 + (3.04 - 6.30i)T \) |
| good | 2 | \( 1 + (0.530 - 0.919i)T + (-2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (-9.99 + 5.76i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 5.56iT - 169T^{2} \) |
| 17 | \( 1 + (-7.82 - 13.5i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-1.06 + 1.84i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-13.0 + 22.5i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 13.3iT - 841T^{2} \) |
| 31 | \( 1 + (-25.3 - 43.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (19.8 + 11.4i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 53.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 38.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-2.72 + 4.71i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-15.5 - 26.8i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (97.9 - 56.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (14.8 - 25.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-23.5 + 13.6i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 79.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-1.70 + 0.981i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-43.8 + 75.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 71.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-106. - 61.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 1.02iT - 9.40e3T^{2} \) |
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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
−14.02258822270382910314901644492, −12.50750136490351815408188587821, −12.02436128550017373463593354467, −10.52575641627043076853862152983, −8.917739261213827738375796858062, −8.386174204869634771199511797587, −7.20862959857878013758417634385, −6.28915922781058128025933060222, −3.71714283654786154020038916265, −2.77631226923327885045307993238,
1.29010157416730295557730040311, 3.33034808235857770858893882388, 4.63259861610294059997721965221, 6.75759102266963141958811377299, 7.81203832222291642951465760050, 9.336677702026518450960967786562, 9.713955128787968756634597051046, 11.14077700598523973265567564167, 12.10469722204224496515358732501, 13.32578692855648074343058059853
