| L(s) = 1 | + (3.04 + 2.21i)2-s − 19.4·3-s + (−35.1 − 108. i)4-s + (86.0 + 264. i)5-s + (−59.1 − 42.9i)6-s + (326. − 236. i)7-s + (281. − 865. i)8-s − 1.80e3·9-s + (−323. + 995. i)10-s + (−1.55e3 + 4.79e3i)11-s + (683. + 2.10e3i)12-s + (1.02e4 + 7.46e3i)13-s + 1.51e3·14-s + (−1.67e3 − 5.14e3i)15-s + (−9.02e3 + 6.55e3i)16-s + (−6.63e3 + 2.04e4i)17-s + ⋯ |
| L(s) = 1 | + (0.269 + 0.195i)2-s − 0.415·3-s + (−0.274 − 0.845i)4-s + (0.307 + 0.947i)5-s + (−0.111 − 0.0812i)6-s + (0.359 − 0.261i)7-s + (0.194 − 0.597i)8-s − 0.827·9-s + (−0.102 + 0.314i)10-s + (−0.352 + 1.08i)11-s + (0.114 + 0.351i)12-s + (1.29 + 0.942i)13-s + 0.147·14-s + (−0.127 − 0.393i)15-s + (−0.550 + 0.400i)16-s + (−0.327 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.689318 + 0.953098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.689318 + 0.953098i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 + (-3.64e5 + 2.49e5i)T \) |
| good | 2 | \( 1 + (-3.04 - 2.21i)T + (39.5 + 121. i)T^{2} \) |
| 3 | \( 1 + 19.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + (-86.0 - 264. i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-326. + 236. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 11 | \( 1 + (1.55e3 - 4.79e3i)T + (-1.57e7 - 1.14e7i)T^{2} \) |
| 13 | \( 1 + (-1.02e4 - 7.46e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (6.63e3 - 2.04e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (3.88e4 - 2.82e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + (1.41e4 + 1.02e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-6.50e4 - 2.00e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-4.35e3 + 1.34e4i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (1.27e5 + 3.92e5i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 43 | \( 1 + (4.24e5 + 3.08e5i)T + (8.39e10 + 2.58e11i)T^{2} \) |
| 47 | \( 1 + (-9.81e5 - 7.13e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (4.08e5 + 1.25e6i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (5.34e5 + 3.88e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (5.35e5 - 3.88e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (3.51e5 + 1.08e6i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (1.57e6 - 4.84e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 - 2.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.24e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.42e6 + 3.94e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-3.80e6 - 1.17e7i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−14.54225946494271914191674703264, −14.23197280517920734718321189553, −12.68104826782084500638588395598, −10.89864439276608712651133464523, −10.47482961398888416658590979943, −8.746126650080616069084897873303, −6.74042233477207114314407873381, −5.86492488805274226978285336136, −4.21068038548101144706593045932, −1.80839296072353035365286784082,
0.48382937113561552192045950554, 2.89300959645813296768644863690, 4.74210359369630398128708043713, 5.93826171559892080902879186722, 8.283985719249400630632357196851, 8.788256810199485683466153737238, 10.97552043403321402404893009237, 11.82106318865194921808146898571, 13.17596641245414079111541235980, 13.68398091683636830607854253644
