| 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 | \( 1 - 2.95e8T + 7.60e17T^{2} \) |
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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
−14.09426929585907282192775928427, −12.67753059841680816925327814996, −11.85070528023922635636277965867, −10.63520203541616778560493748242, −8.434450989779014656249241858779, −6.87634341962146093250011949782, −6.22675890188411410106826212786, −4.88580207559936644419855878751, −3.91799174175269742523707581059, −0.915995072698356937343548731716,
1.18076605054169447374066306096, 2.76657174956922964760335262543, 4.47509612295421501097004364898, 5.34401449252910588252511556986, 6.60408983434693614894038286883, 9.070026755319839417898809741714, 10.73289575576157666411471548275, 11.41927065896478248445697590841, 12.09524119030467605987356074043, 13.13214765231082469627182726102
