L(s) = 1 | + (−15.4 + 1.85i)3-s + 55.8i·5-s − 225. i·7-s + (236. − 57.4i)9-s + 36.9·11-s + 795.·13-s + (−103. − 864. i)15-s − 129. i·17-s + 998. i·19-s + (419. + 3.49e3i)21-s − 3.74e3·23-s + 4.87·25-s + (−3.54e3 + 1.32e3i)27-s + 921. i·29-s − 423. i·31-s + ⋯ |
L(s) = 1 | + (−0.992 + 0.118i)3-s + 0.999i·5-s − 1.74i·7-s + (0.971 − 0.236i)9-s + 0.0920·11-s + 1.30·13-s + (−0.118 − 0.992i)15-s − 0.108i·17-s + 0.634i·19-s + (0.207 + 1.73i)21-s − 1.47·23-s + 0.00156·25-s + (−0.936 + 0.350i)27-s + 0.203i·29-s − 0.0791i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.078943029\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078943029\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.4 - 1.85i)T \) |
good | 5 | \( 1 - 55.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 225. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 36.9T + 1.61e5T^{2} \) |
| 13 | \( 1 - 795.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 129. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 998. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 921. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 423. iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.44e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.08e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.13e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.97e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.56e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.15e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.25e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
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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
−10.41782128999241945092703514107, −9.948813724342443285618568040628, −8.264577418983085357861135857703, −7.13172260776066592127339575223, −6.64545131259025190105602099489, −5.62381568943796513779808241962, −4.13443753736652619246122448829, −3.58610357096297462826297335657, −1.54550054656975866130083785592, −0.37206896844195966584935148081,
1.02808718949820091952996802238, 2.18358017569545166388710066644, 3.98496297852448691661482049921, 5.21167236064050543161271180680, 5.74597061155040051623428486006, 6.64118271052433530705384332617, 8.213901612004015367544910280320, 8.819871926664698106868468191579, 9.739417682605769368002644174105, 10.96585564871937959178858990971