| L(s) = 1 | + (−0.535 + 0.844i)2-s + (1.74 − 1.63i)3-s + (−0.425 − 0.904i)4-s + (2.16 − 0.555i)5-s + (0.447 + 2.34i)6-s + (−0.662 + 0.481i)7-s + (0.992 + 0.125i)8-s + (0.170 − 2.71i)9-s + (−0.692 + 2.12i)10-s + (−0.622 + 0.980i)11-s + (−2.22 − 0.880i)12-s + (0.133 − 2.12i)13-s + (−0.0514 − 0.817i)14-s + (2.86 − 4.51i)15-s + (−0.637 + 0.770i)16-s + (−0.185 + 0.393i)17-s + ⋯ |
| L(s) = 1 | + (−0.378 + 0.597i)2-s + (1.00 − 0.944i)3-s + (−0.212 − 0.452i)4-s + (0.968 − 0.248i)5-s + (0.182 + 0.958i)6-s + (−0.250 + 0.181i)7-s + (0.350 + 0.0443i)8-s + (0.0568 − 0.903i)9-s + (−0.218 + 0.672i)10-s + (−0.187 + 0.295i)11-s + (−0.641 − 0.254i)12-s + (0.0370 − 0.589i)13-s + (−0.0137 − 0.218i)14-s + (0.740 − 1.16i)15-s + (−0.159 + 0.192i)16-s + (−0.0449 + 0.0954i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50268 - 0.229491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50268 - 0.229491i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.535 - 0.844i)T \) |
| 5 | \( 1 + (-2.16 + 0.555i)T \) |
| good | 3 | \( 1 + (-1.74 + 1.63i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (0.662 - 0.481i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (0.622 - 0.980i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (-0.133 + 2.12i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (0.185 - 0.393i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (1.64 + 1.54i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.247 - 0.0636i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (-0.124 - 0.0684i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (3.81 - 8.11i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (-0.518 + 0.626i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (2.75 - 0.707i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (3.78 - 11.6i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-8.82 + 1.11i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (-2.34 + 12.2i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (13.2 + 5.24i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (7.49 + 1.92i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (-5.23 + 2.87i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (3.87 - 0.489i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (5.03 - 1.99i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (12.3 - 11.5i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-6.88 - 6.46i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (4.96 - 1.96i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (-11.2 - 6.18i)T + (51.9 + 81.8i)T^{2} \) |
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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
−12.55913231177314926047536151349, −10.79683606007765207318856720879, −9.747502186936784657021724867345, −8.901324843048366416616597916510, −8.156091001315052954875523668825, −7.12284607867951077015835897730, −6.23317888263863463382417296090, −5.01661771876908717647748288648, −2.90757004732099503089666835421, −1.60614232496968805831940860939,
2.14082642951894791874910599909, 3.28708903472093132605554317470, 4.39114945864720688989636970950, 5.94341448303439872545040680640, 7.40153198620505363980736054489, 8.769716384768634254303613458284, 9.233488413117070122057988402524, 10.16579018669966694001713826182, 10.68482072482791249690562359416, 11.98479413003349804208854983930
