| L(s) = 1 | + (−1.41 + 0.877i)2-s + (−0.273 − 0.961i)3-s + (0.345 − 0.694i)4-s + (0.0873 − 0.0338i)5-s + (1.23 + 1.12i)6-s + (−0.260 + 2.80i)7-s + (−0.188 − 2.03i)8-s + (−0.850 + 0.526i)9-s + (−0.0940 + 0.124i)10-s + (0.0119 − 0.00738i)11-s + (−0.762 − 0.142i)12-s + (0.0682 − 0.736i)13-s + (−2.09 − 4.20i)14-s + (−0.0564 − 0.0747i)15-s + (2.98 + 3.94i)16-s + (−1.93 + 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 + 0.620i)2-s + (−0.157 − 0.555i)3-s + (0.172 − 0.347i)4-s + (0.0390 − 0.0151i)5-s + (0.502 + 0.458i)6-s + (−0.0983 + 1.06i)7-s + (−0.0665 − 0.718i)8-s + (−0.283 + 0.175i)9-s + (−0.0297 + 0.0393i)10-s + (0.00359 − 0.00222i)11-s + (−0.220 − 0.0411i)12-s + (0.0189 − 0.204i)13-s + (−0.559 − 1.12i)14-s + (−0.0145 − 0.0193i)15-s + (0.745 + 0.987i)16-s + (−0.468 + 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.148364 + 0.412570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148364 + 0.412570i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.273 + 0.961i)T \) |
| 103 | \( 1 + (-9.34 + 3.96i)T \) |
| good | 2 | \( 1 + (1.41 - 0.877i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (-0.0873 + 0.0338i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (0.260 - 2.80i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (-0.0119 + 0.00738i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (-0.0682 + 0.736i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (1.93 - 1.76i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.26 - 7.96i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (0.399 + 0.247i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (7.47 - 2.89i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-1.17 - 1.56i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-1.88 - 0.352i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (6.53 + 2.53i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (0.862 - 0.161i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 - 7.03T + 47T^{2} \) |
| 53 | \( 1 + (1.90 - 6.69i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (-0.375 - 4.04i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (-2.07 + 1.89i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (1.45 + 15.6i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (-10.1 - 3.92i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (-9.59 - 3.71i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (-2.45 + 0.950i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (0.0686 - 0.740i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (-1.75 - 3.51i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (7.19 + 6.55i)T + (8.95 + 96.5i)T^{2} \) |
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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
−12.27151923184025246216429882088, −10.99198752165398644969100884275, −9.921610564450119670943240567780, −8.991823388613987399464050152609, −8.262234456842122018441426704241, −7.46123597584903419539410750843, −6.31460255723002989745214844835, −5.59774138440899751472337078274, −3.68160193224937606657259027591, −1.84088577468627205469380761932,
0.44137911264018412329854590912, 2.36521083010503945287175857927, 3.99715587909430616140381952830, 5.10852903146138000626208151524, 6.59355306765776606814335092993, 7.72173692730468774226209355950, 8.850341593908121498215418833148, 9.588074220543177573565113070127, 10.33841583581481331242486288204, 11.14099831747419534924549074575
