| L(s) = 1 | + (0.0324 + 0.360i)2-s + (1.83 − 0.333i)4-s + (0.370 − 0.509i)5-s + (2.04 + 3.42i)7-s + (0.372 + 1.34i)8-s + (0.195 + 0.116i)10-s + (−3.34 + 5.06i)11-s + (−3.55 + 2.34i)13-s + (−1.16 + 0.847i)14-s + (3.02 − 1.13i)16-s + (−0.290 + 0.893i)17-s + (0.441 + 0.462i)19-s + (0.510 − 1.06i)20-s + (−1.93 − 1.03i)22-s + (−1.88 − 8.24i)23-s + ⋯ |
| L(s) = 1 | + (0.0229 + 0.254i)2-s + (0.919 − 0.166i)4-s + (0.165 − 0.227i)5-s + (0.772 + 1.29i)7-s + (0.131 + 0.477i)8-s + (0.0618 + 0.0369i)10-s + (−1.00 + 1.52i)11-s + (−0.985 + 0.650i)13-s + (−0.311 + 0.226i)14-s + (0.756 − 0.283i)16-s + (−0.0704 + 0.216i)17-s + (0.101 + 0.106i)19-s + (0.114 − 0.237i)20-s + (−0.411 − 0.221i)22-s + (−0.392 − 1.71i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55200 + 1.07238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55200 + 1.07238i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (-2.48 + 8.05i)T \) |
| good | 2 | \( 1 + (-0.0324 - 0.360i)T + (-1.96 + 0.357i)T^{2} \) |
| 5 | \( 1 + (-0.370 + 0.509i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-2.04 - 3.42i)T + (-3.31 + 6.16i)T^{2} \) |
| 11 | \( 1 + (3.34 - 5.06i)T + (-4.32 - 10.1i)T^{2} \) |
| 13 | \( 1 + (3.55 - 2.34i)T + (5.10 - 11.9i)T^{2} \) |
| 17 | \( 1 + (0.290 - 0.893i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.441 - 0.462i)T + (-0.852 + 18.9i)T^{2} \) |
| 23 | \( 1 + (1.88 + 8.24i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-2.87 + 1.54i)T + (15.9 - 24.2i)T^{2} \) |
| 31 | \( 1 + (-2.14 + 5.70i)T + (-23.3 - 20.3i)T^{2} \) |
| 37 | \( 1 + (-1.17 + 5.14i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (1.68 - 2.11i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-4.26 + 3.72i)T + (5.77 - 42.6i)T^{2} \) |
| 47 | \( 1 + (0.399 + 2.94i)T + (-45.3 + 12.5i)T^{2} \) |
| 53 | \( 1 + (-5.65 - 1.02i)T + (49.6 + 18.6i)T^{2} \) |
| 59 | \( 1 + (-0.216 - 4.82i)T + (-58.7 + 5.28i)T^{2} \) |
| 61 | \( 1 + (-4.04 + 6.77i)T + (-28.9 - 53.7i)T^{2} \) |
| 67 | \( 1 + (0.301 + 1.65i)T + (-62.7 + 23.5i)T^{2} \) |
| 73 | \( 1 + (14.4 - 1.29i)T + (71.8 - 13.0i)T^{2} \) |
| 79 | \( 1 + (-8.91 + 2.45i)T + (67.8 - 40.5i)T^{2} \) |
| 83 | \( 1 + (-11.4 + 0.514i)T + (82.6 - 7.44i)T^{2} \) |
| 89 | \( 1 + (-1.04 + 5.76i)T + (-83.3 - 31.2i)T^{2} \) |
| 97 | \( 1 + (10.3 - 8.26i)T + (21.5 - 94.5i)T^{2} \) |
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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
−10.68953088970742968843338076325, −9.947735122749524588000407583875, −8.964823988923570746401680798303, −7.953789745687875351737524952481, −7.28330909222907690279859224272, −6.23737600132358146454547117156, −5.24693419348447254262163228732, −4.61107548269917223427049943052, −2.36940331732913694580772232640, −2.14858283535377926432886789919,
1.03334054509471224451233965108, 2.63370917463159877157443470676, 3.48231411711365678476521293080, 4.88769172217142818413116476753, 5.89124578302970300263548419717, 7.05309846071531386996859413027, 7.70210300525839473622827342916, 8.376408873407816949613327284695, 10.03162400634313782692490819636, 10.46528465601251258472062462736
