| L(s) = 1 | + (1.30 − 0.543i)2-s + (1.15 + 1.28i)3-s + (1.40 − 1.42i)4-s + (−0.183 − 0.209i)5-s + (2.21 + 1.05i)6-s + (1.21 + 0.159i)7-s + (1.06 − 2.61i)8-s + (−0.319 + 2.98i)9-s + (−0.354 − 0.173i)10-s + (−0.510 − 1.03i)11-s + (3.45 + 0.169i)12-s + (−0.0481 − 0.141i)13-s + (1.67 − 0.451i)14-s + (0.0572 − 0.479i)15-s + (−0.0334 − 3.99i)16-s + (2.67 + 2.67i)17-s + ⋯ |
| L(s) = 1 | + (0.923 − 0.384i)2-s + (0.668 + 0.743i)3-s + (0.704 − 0.710i)4-s + (−0.0822 − 0.0937i)5-s + (0.903 + 0.429i)6-s + (0.458 + 0.0603i)7-s + (0.376 − 0.926i)8-s + (−0.106 + 0.994i)9-s + (−0.111 − 0.0549i)10-s + (−0.153 − 0.311i)11-s + (0.998 + 0.0490i)12-s + (−0.0133 − 0.0393i)13-s + (0.446 − 0.120i)14-s + (0.0147 − 0.123i)15-s + (−0.00837 − 0.999i)16-s + (0.648 + 0.648i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.15240 - 0.165508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.15240 - 0.165508i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 0.543i)T \) |
| 3 | \( 1 + (-1.15 - 1.28i)T \) |
| good | 5 | \( 1 + (0.183 + 0.209i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-1.21 - 0.159i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (0.510 + 1.03i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.0481 + 0.141i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-2.67 - 2.67i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.709 + 3.56i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.854 - 6.49i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (5.01 - 0.328i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (7.19 - 4.15i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.345 - 1.73i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.636 + 4.83i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-5.19 + 2.56i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (7.62 - 2.04i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.478 - 0.715i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-5.66 - 6.45i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.723 + 11.0i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-1.14 - 0.565i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (1.43 + 3.46i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (1.96 - 4.73i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (0.0723 + 0.270i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-0.835 - 0.732i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (16.2 - 6.74i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-7.35 - 4.24i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.88037788064837867965884399398, −9.966830099626529807277160456489, −9.130364967757855934840606585418, −8.067295761106794627500580325911, −7.13763195684494315062125810424, −5.68025485451800137090336985836, −5.01527275590618113833419180597, −3.92437021363297209602976155147, −3.11327955945123441352539304499, −1.79960525672504898841649150419,
1.78377202963863188829882691106, 2.97878735095466047686392392532, 3.99976204458748558972315018531, 5.22934520617215579211388344219, 6.21656673298194847891769751321, 7.29751353189576914661472486480, 7.71839126343814919435770345282, 8.671454146017057762234407174723, 9.771786890014518626174731667920, 11.11049970599590762426033754782
