| L(s) = 1 | + (3.05 + 3.05i)2-s + (−299. + 299. i)3-s − 4.07e3i·4-s + (−1.56e4 + 583. i)5-s − 1.82e3·6-s + (−1.39e5 − 1.39e5i)7-s + (2.49e4 − 2.49e4i)8-s + 3.51e5i·9-s + (−4.94e4 − 4.58e4i)10-s + 1.90e6·11-s + (1.22e6 + 1.22e6i)12-s + (−1.77e6 + 1.77e6i)13-s − 8.53e5i·14-s + (4.50e6 − 4.85e6i)15-s − 1.65e7·16-s + (−1.14e7 − 1.14e7i)17-s + ⋯ |
| L(s) = 1 | + (0.0476 + 0.0476i)2-s + (−0.411 + 0.411i)3-s − 0.995i·4-s + (−0.999 + 0.0373i)5-s − 0.0392·6-s + (−1.18 − 1.18i)7-s + (0.0951 − 0.0951i)8-s + 0.661i·9-s + (−0.0494 − 0.0458i)10-s + 1.07·11-s + (0.409 + 0.409i)12-s + (−0.366 + 0.366i)13-s − 0.113i·14-s + (0.395 − 0.426i)15-s − 0.986·16-s + (−0.473 − 0.473i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.132074 - 0.433931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132074 - 0.433931i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.56e4 - 583. i)T \) |
| good | 2 | \( 1 + (-3.05 - 3.05i)T + 4.09e3iT^{2} \) |
| 3 | \( 1 + (299. - 299. i)T - 5.31e5iT^{2} \) |
| 7 | \( 1 + (1.39e5 + 1.39e5i)T + 1.38e10iT^{2} \) |
| 11 | \( 1 - 1.90e6T + 3.13e12T^{2} \) |
| 13 | \( 1 + (1.77e6 - 1.77e6i)T - 2.32e13iT^{2} \) |
| 17 | \( 1 + (1.14e7 + 1.14e7i)T + 5.82e14iT^{2} \) |
| 19 | \( 1 + 2.79e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + (3.91e7 - 3.91e7i)T - 2.19e16iT^{2} \) |
| 29 | \( 1 + 8.54e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 - 5.03e8T + 7.87e17T^{2} \) |
| 37 | \( 1 + (1.34e9 + 1.34e9i)T + 6.58e18iT^{2} \) |
| 41 | \( 1 + 2.34e9T + 2.25e19T^{2} \) |
| 43 | \( 1 + (4.61e9 - 4.61e9i)T - 3.99e19iT^{2} \) |
| 47 | \( 1 + (-2.02e8 - 2.02e8i)T + 1.16e20iT^{2} \) |
| 53 | \( 1 + (-1.74e10 + 1.74e10i)T - 4.91e20iT^{2} \) |
| 59 | \( 1 + 1.23e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 + 6.84e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + (-4.44e10 - 4.44e10i)T + 8.18e21iT^{2} \) |
| 71 | \( 1 - 2.05e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (2.31e10 - 2.31e10i)T - 2.29e22iT^{2} \) |
| 79 | \( 1 + 7.81e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (3.18e11 - 3.18e11i)T - 1.06e23iT^{2} \) |
| 89 | \( 1 + 9.33e10iT - 2.46e23T^{2} \) |
| 97 | \( 1 + (9.38e11 + 9.38e11i)T + 6.93e23iT^{2} \) |
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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
−19.83476529974870660171255513362, −19.33468591630485906591742230551, −16.76985389194391676464745839556, −15.61197946211226520168493980888, −13.76009342102866049764560432982, −11.31807489591851844042676548314, −9.842036550969553972534579850728, −6.81935973779865619677581615222, −4.31549136902700333800674775640, −0.29991240638531929653460436243,
3.44702251774419837339547630938, 6.68191866999439714740911566118, 8.808103077001970462287547392899, 11.97410771697697581046819478211, 12.52067943223938082458338776562, 15.36180712513035033996220906199, 16.80518977917580087672567289203, 18.48871269402600294229415797125, 19.95839210416735839675864059458, 21.96989365176697246528097927088
