| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−1.67 + 0.457i)3-s + (−0.959 + 0.281i)4-s + (1.26 − 1.84i)5-s + (0.690 + 1.58i)6-s + (−1.66 + 1.07i)7-s + (0.415 + 0.909i)8-s + (2.58 − 1.52i)9-s + (−2.00 − 0.988i)10-s + (−0.163 + 1.14i)11-s + (1.47 − 0.909i)12-s + (3.31 − 5.15i)13-s + (1.29 + 1.49i)14-s + (−1.26 + 3.65i)15-s + (0.841 − 0.540i)16-s + (−0.599 + 2.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.964 + 0.264i)3-s + (−0.479 + 0.140i)4-s + (0.565 − 0.824i)5-s + (0.281 + 0.648i)6-s + (−0.629 + 0.404i)7-s + (0.146 + 0.321i)8-s + (0.860 − 0.509i)9-s + (−0.634 − 0.312i)10-s + (−0.0494 + 0.343i)11-s + (0.425 − 0.262i)12-s + (0.919 − 1.43i)13-s + (0.346 + 0.399i)14-s + (−0.327 + 0.944i)15-s + (0.210 − 0.135i)16-s + (−0.145 + 0.495i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.156069 - 0.671226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.156069 - 0.671226i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (1.67 - 0.457i)T \) |
| 5 | \( 1 + (-1.26 + 1.84i)T \) |
| 23 | \( 1 + (4.69 - 0.979i)T \) |
| good | 7 | \( 1 + (1.66 - 1.07i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.163 - 1.14i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-3.31 + 5.15i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.599 - 2.04i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.290 + 0.988i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.730 + 2.48i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.50 + 5.48i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (5.56 + 6.42i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (2.34 + 2.03i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-5.08 + 11.1i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 + (-2.42 - 3.76i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (3.10 - 4.83i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (9.92 - 4.53i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.884 + 6.15i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (12.3 - 1.77i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.762 - 2.59i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-6.33 + 9.85i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-12.6 + 10.9i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.37 + 9.58i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (4.00 - 4.62i)T + (-13.8 - 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32155075785841739771444073531, −9.411076190582975775682723278971, −8.722416838057173229961176443911, −7.56481102322398185687787860292, −5.95072472198508699050101975240, −5.77281171183646126682818098271, −4.54714370318343175793085006496, −3.53004663062453957080960653724, −1.91661789677950594318079446783, −0.43426130960925899683990550220,
1.60642434648427896152518732291, 3.45162790724895520079355700288, 4.65079876610841097500553432456, 5.80532670625979533263115479022, 6.66398774580039749724281940889, 6.75976854591140021160210905611, 8.030280191464524832682682559885, 9.239339107058965028417922991249, 10.00391014684186266486140661542, 10.78578266483826301862230538158
