| L(s) = 1 | + (27.1 − 25.1i)2-s + (−160. + 24.2i)3-s + (63.8 − 852. i)4-s + (−112. + 287. i)5-s + (−3.74e3 + 4.69e3i)6-s + (3.13e3 + 5.52e3i)7-s + (−7.90e3 − 9.91e3i)8-s + (6.41e3 − 1.97e3i)9-s + (4.16e3 + 1.06e4i)10-s + (6.61e4 + 2.04e4i)11-s + (1.03e4 + 1.38e5i)12-s + (−7.67e3 − 3.36e4i)13-s + (2.23e5 + 7.07e4i)14-s + (1.11e4 − 4.88e4i)15-s + (−3.06e4 − 4.62e3i)16-s + (−4.77e4 + 3.25e4i)17-s + ⋯ |
| L(s) = 1 | + (1.19 − 1.11i)2-s + (−1.14 + 0.172i)3-s + (0.124 − 1.66i)4-s + (−0.0806 + 0.205i)5-s + (−1.17 + 1.47i)6-s + (0.494 + 0.869i)7-s + (−0.682 − 0.855i)8-s + (0.325 − 0.100i)9-s + (0.131 + 0.335i)10-s + (1.36 + 0.420i)11-s + (0.144 + 1.92i)12-s + (−0.0744 − 0.326i)13-s + (1.55 + 0.491i)14-s + (0.0568 − 0.249i)15-s + (−0.116 − 0.0176i)16-s + (−0.138 + 0.0945i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.58518 - 0.970874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.58518 - 0.970874i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-3.13e3 - 5.52e3i)T \) |
| good | 2 | \( 1 + (-27.1 + 25.1i)T + (38.2 - 510. i)T^{2} \) |
| 3 | \( 1 + (160. - 24.2i)T + (1.88e4 - 5.80e3i)T^{2} \) |
| 5 | \( 1 + (112. - 287. i)T + (-1.43e6 - 1.32e6i)T^{2} \) |
| 11 | \( 1 + (-6.61e4 - 2.04e4i)T + (1.94e9 + 1.32e9i)T^{2} \) |
| 13 | \( 1 + (7.67e3 + 3.36e4i)T + (-9.55e9 + 4.60e9i)T^{2} \) |
| 17 | \( 1 + (4.77e4 - 3.25e4i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (4.82e5 + 8.35e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.52e6 - 1.03e6i)T + (6.58e11 + 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-2.91e6 - 1.40e6i)T + (9.04e12 + 1.13e13i)T^{2} \) |
| 31 | \( 1 + (-2.09e5 + 3.63e5i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (2.36e5 + 3.15e6i)T + (-1.28e14 + 1.93e13i)T^{2} \) |
| 41 | \( 1 + (-1.06e7 - 1.34e7i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 43 | \( 1 + (1.53e7 - 1.91e7i)T + (-1.11e14 - 4.89e14i)T^{2} \) |
| 47 | \( 1 + (-3.12e7 + 2.90e7i)T + (8.36e13 - 1.11e15i)T^{2} \) |
| 53 | \( 1 + (2.55e6 - 3.40e7i)T + (-3.26e15 - 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-3.09e7 - 7.87e7i)T + (-6.35e15 + 5.89e15i)T^{2} \) |
| 61 | \( 1 + (-5.10e6 - 6.80e7i)T + (-1.15e16 + 1.74e15i)T^{2} \) |
| 67 | \( 1 + (-4.39e6 + 7.60e6i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.22e8 + 5.90e7i)T + (2.85e16 - 3.58e16i)T^{2} \) |
| 73 | \( 1 + (-1.62e6 - 1.51e6i)T + (4.39e15 + 5.87e16i)T^{2} \) |
| 79 | \( 1 + (1.89e8 + 3.27e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (1.14e8 - 5.00e8i)T + (-1.68e17 - 8.11e16i)T^{2} \) |
| 89 | \( 1 + (-8.21e8 + 2.53e8i)T + (2.89e17 - 1.97e17i)T^{2} \) |
| 97 | \( 1 - 2.95e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.13214765231082469627182726102, −12.09524119030467605987356074043, −11.41927065896478248445697590841, −10.73289575576157666411471548275, −9.070026755319839417898809741714, −6.60408983434693614894038286883, −5.34401449252910588252511556986, −4.47509612295421501097004364898, −2.76657174956922964760335262543, −1.18076605054169447374066306096,
0.915995072698356937343548731716, 3.91799174175269742523707581059, 4.88580207559936644419855878751, 6.22675890188411410106826212786, 6.87634341962146093250011949782, 8.434450989779014656249241858779, 10.63520203541616778560493748242, 11.85070528023922635636277965867, 12.67753059841680816925327814996, 14.09426929585907282192775928427
