| L(s) = 1 | + (94.8 − 164. i)2-s + (−9.42e3 + 5.43e3i)3-s + (1.47e4 + 2.56e4i)4-s + (−4.77e5 − 2.75e5i)5-s + 2.06e6i·6-s + (5.57e6 − 1.45e6i)7-s + 1.80e7·8-s + (3.76e7 − 6.52e7i)9-s + (−9.04e7 + 5.22e7i)10-s + (7.76e7 + 1.34e8i)11-s + (−2.78e8 − 1.60e8i)12-s − 1.23e9i·13-s + (2.90e8 − 1.05e9i)14-s + 5.99e9·15-s + (7.41e8 − 1.28e9i)16-s + (−1.42e9 + 8.22e8i)17-s + ⋯ |
| L(s) = 1 | + (0.370 − 0.641i)2-s + (−1.43 + 0.829i)3-s + (0.225 + 0.390i)4-s + (−1.22 − 0.705i)5-s + 1.22i·6-s + (0.967 − 0.252i)7-s + 1.07·8-s + (0.874 − 1.51i)9-s + (−0.904 + 0.522i)10-s + (0.362 + 0.627i)11-s + (−0.647 − 0.373i)12-s − 1.51i·13-s + (0.196 − 0.714i)14-s + 2.33·15-s + (0.172 − 0.299i)16-s + (−0.204 + 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(1.13586 - 0.562117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13586 - 0.562117i\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-5.57e6 + 1.45e6i)T \) |
| good | 2 | \( 1 + (-94.8 + 164. i)T + (-3.27e4 - 5.67e4i)T^{2} \) |
| 3 | \( 1 + (9.42e3 - 5.43e3i)T + (2.15e7 - 3.72e7i)T^{2} \) |
| 5 | \( 1 + (4.77e5 + 2.75e5i)T + (7.62e10 + 1.32e11i)T^{2} \) |
| 11 | \( 1 + (-7.76e7 - 1.34e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 13 | \( 1 + 1.23e9iT - 6.65e17T^{2} \) |
| 17 | \( 1 + (1.42e9 - 8.22e8i)T + (2.43e19 - 4.21e19i)T^{2} \) |
| 19 | \( 1 + (-1.62e10 - 9.38e9i)T + (1.44e20 + 2.49e20i)T^{2} \) |
| 23 | \( 1 + (-3.57e10 + 6.18e10i)T + (-3.06e21 - 5.31e21i)T^{2} \) |
| 29 | \( 1 + 1.65e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + (-7.81e11 + 4.51e11i)T + (3.63e23 - 6.29e23i)T^{2} \) |
| 37 | \( 1 + (-1.77e12 + 3.06e12i)T + (-6.16e24 - 1.06e25i)T^{2} \) |
| 41 | \( 1 + 7.99e12iT - 6.37e25T^{2} \) |
| 43 | \( 1 + 8.81e11T + 1.36e26T^{2} \) |
| 47 | \( 1 + (3.12e12 + 1.80e12i)T + (2.83e26 + 4.91e26i)T^{2} \) |
| 53 | \( 1 + (1.65e13 + 2.86e13i)T + (-1.93e27 + 3.35e27i)T^{2} \) |
| 59 | \( 1 + (1.25e14 - 7.25e13i)T + (1.07e28 - 1.86e28i)T^{2} \) |
| 61 | \( 1 + (-3.49e13 - 2.01e13i)T + (1.83e28 + 3.18e28i)T^{2} \) |
| 67 | \( 1 + (6.43e12 + 1.11e13i)T + (-8.24e28 + 1.42e29i)T^{2} \) |
| 71 | \( 1 - 3.62e13T + 4.16e29T^{2} \) |
| 73 | \( 1 + (-7.23e14 + 4.17e14i)T + (3.25e29 - 5.63e29i)T^{2} \) |
| 79 | \( 1 + (-4.69e14 + 8.13e14i)T + (-1.15e30 - 1.99e30i)T^{2} \) |
| 83 | \( 1 - 7.02e14iT - 5.07e30T^{2} \) |
| 89 | \( 1 + (-7.22e14 - 4.17e14i)T + (7.74e30 + 1.34e31i)T^{2} \) |
| 97 | \( 1 - 1.18e16iT - 6.14e31T^{2} \) |
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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
−17.61568664397012571117526931568, −16.54384928963359737050046202800, −15.37375196236541308720075544250, −12.47774775292852862662869015368, −11.64856050932928060602938477937, −10.58794525795879927053823210893, −7.77940838724113314894115565431, −4.99465737227170849256600033215, −3.97277961391613305846321219062, −0.75290976985056460613235451497,
1.23681808935460301181421381162, 4.81699241727420365166700196155, 6.45949917317764531868064485726, 7.49336546786245419678108355685, 11.35595981284360774351687878787, 11.49618917282917799467869804252, 13.99759695543433142491455177974, 15.53772204097662825854946170885, 16.81574973012149240724394970921, 18.43509198686728728724359124158
