| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.266 + 1.50i)3-s + (0.766 + 0.642i)4-s + (1.53 − 1.28i)5-s + (0.266 − 1.50i)6-s + (−1.34 + 2.33i)7-s + (−0.500 − 0.866i)8-s + (0.613 − 0.223i)9-s + (−1.87 + 0.684i)10-s + (−1.59 − 2.75i)11-s + (−0.766 + 1.32i)12-s + (−1 + 5.67i)13-s + (2.06 − 1.73i)14-s + (2.34 + 1.96i)15-s + (0.173 + 0.984i)16-s + (6.12 + 2.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.153 + 0.871i)3-s + (0.383 + 0.321i)4-s + (0.685 − 0.574i)5-s + (0.108 − 0.615i)6-s + (−0.509 + 0.882i)7-s + (−0.176 − 0.306i)8-s + (0.204 − 0.0744i)9-s + (−0.594 + 0.216i)10-s + (−0.480 − 0.831i)11-s + (−0.221 + 0.383i)12-s + (−0.277 + 1.57i)13-s + (0.551 − 0.462i)14-s + (0.606 + 0.508i)15-s + (0.0434 + 0.246i)16-s + (1.48 + 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.978127 + 0.725114i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.978127 + 0.725114i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-0.266 - 1.50i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-1.53 + 1.28i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.34 - 2.33i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 5.67i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-6.12 - 2.22i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (0.532 + 0.446i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.65 + 0.965i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.22 - 2.12i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 + (0.0603 + 0.342i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.64 + 3.89i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (7.41 - 2.69i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-6.29 - 5.27i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.539 + 0.196i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.24 - 1.88i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.65 + 1.69i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.47 - 5.43i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.73 - 15.5i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.57 + 8.92i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (4.23 - 7.34i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.34 + 7.62i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.326 - 0.118i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.19739152860052591728593192814, −9.720195693595848278658729150429, −9.009564964864386475932623830645, −8.497374082531903891207468191717, −7.16739078286117126837260579469, −6.02802893198488501701854244600, −5.24984099484663041267541216456, −3.98227878873605519833636363514, −2.89280975185635666689372299043, −1.53914989343845511705448426703,
0.819432184242721885958086148150, 2.21421028075093354326513629675, 3.27297525152196898814833051263, 5.05143535554750038411599306496, 6.08811414288198842781279627997, 7.02230927505796553437290052369, 7.53477776490150980076377605836, 8.158616114981115021119230289213, 9.723269805350108844257236591948, 10.13397952299474398859900670740
