| L(s) = 1 | + (0.147 + 1.02i)3-s + (0.959 + 0.281i)5-s + (−0.636 + 0.734i)7-s + (1.84 − 0.540i)9-s + (−4.49 + 2.88i)11-s + (3.87 + 4.47i)13-s + (−0.147 + 1.02i)15-s + (1.45 − 3.18i)17-s + (1.27 + 2.79i)19-s + (−0.850 − 0.546i)21-s + (0.107 − 4.79i)23-s + (0.841 + 0.540i)25-s + (2.12 + 4.65i)27-s + (−4.38 + 9.60i)29-s + (0.341 − 2.37i)31-s + ⋯ |
| L(s) = 1 | + (0.0854 + 0.594i)3-s + (0.429 + 0.125i)5-s + (−0.240 + 0.277i)7-s + (0.613 − 0.180i)9-s + (−1.35 + 0.870i)11-s + (1.07 + 1.24i)13-s + (−0.0382 + 0.265i)15-s + (0.353 − 0.773i)17-s + (0.293 + 0.642i)19-s + (−0.185 − 0.119i)21-s + (0.0224 − 0.999i)23-s + (0.168 + 0.108i)25-s + (0.408 + 0.895i)27-s + (−0.814 + 1.78i)29-s + (0.0612 − 0.426i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19380 + 0.910609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19380 + 0.910609i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-0.107 + 4.79i)T \) |
| good | 3 | \( 1 + (-0.147 - 1.02i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (0.636 - 0.734i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.49 - 2.88i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.87 - 4.47i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.45 + 3.18i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.27 - 2.79i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (4.38 - 9.60i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.341 + 2.37i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (5.02 - 1.47i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-4.51 - 1.32i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.336 + 2.34i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 + (-8.54 + 9.86i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (2.73 + 3.15i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.72 + 11.9i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (5.08 + 3.27i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (10.5 + 6.75i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.92 + 6.40i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-10.7 - 12.4i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.70 + 1.38i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (2.10 + 14.6i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.25 - 0.661i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.97144479435047173767796452950, −10.30194481938439389107611515374, −9.511478131733207611333167804056, −8.809683964557188062242970187741, −7.49259726644943116828798484889, −6.64849994112483180264348104585, −5.43305725045996859756349420445, −4.51568436238989229656108317230, −3.31329233758319351696480600547, −1.87845167489689912320503980085,
1.00406993099286466988974370735, 2.60750933204653686403715020619, 3.84204322974902291129111763922, 5.46607185548003919174009181331, 6.01107015591472703090585383655, 7.40901044272519204313601136749, 7.969341529329803436653610594313, 8.956100627230224251960295582765, 10.31808936996282641712363369932, 10.55984937261848837781172953527
