| L(s) = 1 | + (117. − 203. i)5-s + (881. + 215. i)7-s + (−3.93e3 − 6.81e3i)11-s − 5.76e3·13-s + (−3.57e3 − 6.19e3i)17-s + (−1.99e4 + 3.45e4i)19-s + (−2.82e4 + 4.89e4i)23-s + (1.14e4 + 1.98e4i)25-s − 1.30e5·29-s + (−2.57e4 − 4.46e4i)31-s + (1.47e5 − 1.54e5i)35-s + (1.39e5 − 2.42e5i)37-s − 3.45e5·41-s − 1.11e5·43-s + (4.25e5 − 7.36e5i)47-s + ⋯ |
| L(s) = 1 | + (0.420 − 0.728i)5-s + (0.971 + 0.237i)7-s + (−0.891 − 1.54i)11-s − 0.727·13-s + (−0.176 − 0.305i)17-s + (−0.667 + 1.15i)19-s + (−0.484 + 0.839i)23-s + (0.146 + 0.253i)25-s − 0.990·29-s + (−0.155 − 0.269i)31-s + (0.581 − 0.607i)35-s + (0.453 − 0.785i)37-s − 0.783·41-s − 0.214·43-s + (0.597 − 1.03i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1516378576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1516378576\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-881. - 215. i)T \) |
| good | 5 | \( 1 + (-117. + 203. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (3.93e3 + 6.81e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 5.76e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (3.57e3 + 6.19e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.99e4 - 3.45e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.82e4 - 4.89e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 1.30e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (2.57e4 + 4.46e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.39e5 + 2.42e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 3.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.11e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-4.25e5 + 7.36e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-5.99e5 - 1.03e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-8.56e5 - 1.48e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.22e6 - 2.12e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (3.13e5 + 5.43e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.97e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-4.76e5 - 8.25e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.95e6 - 5.11e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 9.78e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.59e6 + 4.48e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 4.87e6T + 8.07e13T^{2} \) |
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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
−11.18113777015618226051532709401, −10.27202876824013416073576889870, −9.081982965957419557788070995640, −8.321817644424015648090262624135, −7.47855317257205127649281441438, −5.70683776270324325682089842217, −5.36384693780803195449056580830, −3.97256887003308728744571204492, −2.45245956782797546700154780010, −1.31604002762442467840688181910,
0.03238511865991817837331678201, 1.89495337260695133117491748178, 2.57696092918672017478980553854, 4.39212222242611973390665114244, 5.11015369963492658482735783177, 6.61240954457939567031324664080, 7.38046669574600155043499019990, 8.350197861718340390987438685907, 9.700364582741475234360677633013, 10.43555604966542595455967824593
