| L(s) = 1 | + (32 − 55.4i)2-s + (194. + 336. i)3-s + (−2.04e3 − 3.54e3i)4-s + (−1.29e4 + 2.24e4i)5-s + 2.48e4·6-s + (2.86e5 − 1.21e5i)7-s − 2.62e5·8-s + (7.21e5 − 1.24e6i)9-s + (8.28e5 + 1.43e6i)10-s + (1.12e5 + 1.95e5i)11-s + (7.96e5 − 1.37e6i)12-s + 2.48e7·13-s + (2.42e6 − 1.97e7i)14-s − 1.00e7·15-s + (−8.38e6 + 1.45e7i)16-s + (−6.10e7 − 1.05e8i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.153 + 0.266i)3-s + (−0.249 − 0.433i)4-s + (−0.370 + 0.642i)5-s + 0.217·6-s + (0.920 − 0.390i)7-s − 0.353·8-s + (0.452 − 0.783i)9-s + (0.262 + 0.453i)10-s + (0.0191 + 0.0332i)11-s + (0.0769 − 0.133i)12-s + 1.42·13-s + (0.0861 − 0.701i)14-s − 0.228·15-s + (−0.125 + 0.216i)16-s + (−0.613 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.06947 - 1.27729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06947 - 1.27729i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-32 + 55.4i)T \) |
| 7 | \( 1 + (-2.86e5 + 1.21e5i)T \) |
| good | 3 | \( 1 + (-194. - 336. i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + (1.29e4 - 2.24e4i)T + (-6.10e8 - 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-1.12e5 - 1.95e5i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 13 | \( 1 - 2.48e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (6.10e7 + 1.05e8i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-1.07e8 + 1.85e8i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-4.90e8 + 8.50e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + 2.47e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-2.44e9 - 4.22e9i)T + (-1.22e19 + 2.11e19i)T^{2} \) |
| 37 | \( 1 + (1.35e10 - 2.33e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 - 3.27e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.78e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (7.58e9 - 1.31e10i)T + (-2.73e21 - 4.72e21i)T^{2} \) |
| 53 | \( 1 + (1.96e10 + 3.40e10i)T + (-1.30e22 + 2.25e22i)T^{2} \) |
| 59 | \( 1 + (7.22e10 + 1.25e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (7.88e10 - 1.36e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-3.83e11 - 6.64e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 + 1.73e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + (7.94e11 + 1.37e12i)T + (-8.35e23 + 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-1.43e11 + 2.47e11i)T + (-2.33e24 - 4.04e24i)T^{2} \) |
| 83 | \( 1 + 9.41e10T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-1.97e12 + 3.42e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + 1.04e13T + 6.73e25T^{2} \) |
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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
−15.75304208778156599185402469098, −14.65856472534802210394893386072, −13.39984075391755388116824227086, −11.59979965340693629342834122766, −10.68565914071146872393429725985, −8.913413082137581592243520079741, −6.87623821730908310521154746176, −4.60824578979732840832767291432, −3.16670027599768625710971812208, −1.03825390696610814602550554896,
1.54506551789164061948829893906, 4.12014752295418179011544794237, 5.65867954484030400555419836781, 7.71124983541916962398483364255, 8.693247588796120496000553062061, 11.11023730110394110834138892741, 12.69181882181899461931629559346, 13.82932453069745832583434299064, 15.32988010032230673071583762087, 16.37331418714476688910860234007
