| L(s) = 1 | + (−0.568 − 0.822i)2-s + (1.79 + 0.441i)3-s + (−0.354 + 0.935i)4-s + (−0.748 + 0.663i)5-s + (−0.654 − 1.72i)6-s + (−1.88 − 0.464i)7-s + (0.970 − 0.239i)8-s + (0.360 + 0.189i)9-s + (0.970 + 0.239i)10-s + (−0.371 − 0.329i)11-s + (−1.04 + 1.51i)12-s + (−2.41 + 6.36i)13-s + (0.689 + 1.81i)14-s + (−1.63 + 0.857i)15-s + (−0.748 − 0.663i)16-s + (−2.31 + 6.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.401 − 0.581i)2-s + (1.03 + 0.255i)3-s + (−0.177 + 0.467i)4-s + (−0.334 + 0.296i)5-s + (−0.267 − 0.704i)6-s + (−0.713 − 0.175i)7-s + (0.343 − 0.0846i)8-s + (0.120 + 0.0631i)9-s + (0.307 + 0.0756i)10-s + (−0.112 − 0.0993i)11-s + (−0.302 + 0.438i)12-s + (−0.670 + 1.76i)13-s + (0.184 + 0.485i)14-s + (−0.422 + 0.221i)15-s + (−0.187 − 0.165i)16-s + (−0.562 + 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0773 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0773 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.734192 + 0.679431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734192 + 0.679431i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.568 + 0.822i)T \) |
| 5 | \( 1 + (0.748 - 0.663i)T \) |
| 79 | \( 1 + (8.84 - 0.868i)T \) |
| good | 3 | \( 1 + (-1.79 - 0.441i)T + (2.65 + 1.39i)T^{2} \) |
| 7 | \( 1 + (1.88 + 0.464i)T + (6.19 + 3.25i)T^{2} \) |
| 11 | \( 1 + (0.371 + 0.329i)T + (1.32 + 10.9i)T^{2} \) |
| 13 | \( 1 + (2.41 - 6.36i)T + (-9.73 - 8.62i)T^{2} \) |
| 17 | \( 1 + (2.31 - 6.11i)T + (-12.7 - 11.2i)T^{2} \) |
| 19 | \( 1 + (-0.123 - 1.02i)T + (-18.4 + 4.54i)T^{2} \) |
| 23 | \( 1 - 8.42T + 23T^{2} \) |
| 29 | \( 1 + (1.80 - 0.945i)T + (16.4 - 23.8i)T^{2} \) |
| 31 | \( 1 + (-2.33 - 3.38i)T + (-10.9 + 28.9i)T^{2} \) |
| 37 | \( 1 + (-0.358 - 2.95i)T + (-35.9 + 8.85i)T^{2} \) |
| 41 | \( 1 + (8.28 - 7.33i)T + (4.94 - 40.7i)T^{2} \) |
| 43 | \( 1 + (0.877 - 0.777i)T + (5.18 - 42.6i)T^{2} \) |
| 47 | \( 1 + (-0.422 + 3.48i)T + (-45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (-4.98 + 1.22i)T + (46.9 - 24.6i)T^{2} \) |
| 59 | \( 1 + (-2.69 - 7.11i)T + (-44.1 + 39.1i)T^{2} \) |
| 61 | \( 1 + (1.42 + 11.7i)T + (-59.2 + 14.5i)T^{2} \) |
| 67 | \( 1 + (-1.66 + 2.41i)T + (-23.7 - 62.6i)T^{2} \) |
| 71 | \( 1 + (3.76 - 0.927i)T + (62.8 - 32.9i)T^{2} \) |
| 73 | \( 1 + (-1.48 - 3.92i)T + (-54.6 + 48.4i)T^{2} \) |
| 83 | \( 1 + (-1.10 + 2.90i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (-6.42 - 1.58i)T + (78.8 + 41.3i)T^{2} \) |
| 97 | \( 1 + (1.33 + 10.9i)T + (-94.1 + 23.2i)T^{2} \) |
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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
−10.29506705252620688200904416274, −9.580473578766627511550307959764, −8.851389038067243663294476180372, −8.298174015467770989497035986097, −7.11287385866601119486454646197, −6.45980419686925365917302137132, −4.68622210289243469549558712505, −3.72313655936506247331355133977, −3.00020783627879924672607037434, −1.85125591688655225158458279915,
0.48275710573015605660619825507, 2.54443528071593243179924519197, 3.25454535272391414286279601362, 4.86056455860012482006501877689, 5.63320171792985321109664471598, 7.06291357011779124253639854276, 7.48071663698224825225812398166, 8.408759011443101768243329749989, 9.073787778705179005163453926971, 9.720087633569306751154055657950
