| L(s) = 1 | + (−0.935 + 0.354i)2-s + (−0.229 − 0.437i)3-s + (0.748 − 0.663i)4-s + (−1.31 + 1.80i)5-s + (0.370 + 0.327i)6-s + (1.16 + 2.21i)7-s + (−0.464 + 0.885i)8-s + (1.56 − 2.26i)9-s + (0.591 − 2.15i)10-s + (0.397 − 3.27i)11-s + (−0.462 − 0.175i)12-s + (1.19 + 1.35i)13-s + (−1.87 − 1.66i)14-s + (1.09 + 0.161i)15-s + (0.120 − 0.992i)16-s + (0.108 + 0.122i)17-s + ⋯ |
| L(s) = 1 | + (−0.661 + 0.250i)2-s + (−0.132 − 0.252i)3-s + (0.374 − 0.331i)4-s + (−0.589 + 0.807i)5-s + (0.151 + 0.133i)6-s + (0.439 + 0.838i)7-s + (−0.164 + 0.313i)8-s + (0.521 − 0.755i)9-s + (0.187 − 0.681i)10-s + (0.119 − 0.988i)11-s + (−0.133 − 0.0506i)12-s + (0.332 + 0.375i)13-s + (−0.500 − 0.443i)14-s + (0.282 + 0.0417i)15-s + (0.0301 − 0.248i)16-s + (0.0262 + 0.0295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.930125 + 0.462656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930125 + 0.462656i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.935 - 0.354i)T \) |
| 5 | \( 1 + (1.31 - 1.80i)T \) |
| 79 | \( 1 + (-0.408 - 8.87i)T \) |
| good | 3 | \( 1 + (0.229 + 0.437i)T + (-1.70 + 2.46i)T^{2} \) |
| 7 | \( 1 + (-1.16 - 2.21i)T + (-3.97 + 5.76i)T^{2} \) |
| 11 | \( 1 + (-0.397 + 3.27i)T + (-10.6 - 2.63i)T^{2} \) |
| 13 | \( 1 + (-1.19 - 1.35i)T + (-1.56 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.108 - 0.122i)T + (-2.04 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.43 - 0.354i)T + (16.8 + 8.82i)T^{2} \) |
| 23 | \( 1 - 0.958iT - 23T^{2} \) |
| 29 | \( 1 + (-1.29 - 1.88i)T + (-10.2 + 27.1i)T^{2} \) |
| 31 | \( 1 + (-3.16 - 8.33i)T + (-23.2 + 20.5i)T^{2} \) |
| 37 | \( 1 + (0.821 - 3.33i)T + (-32.7 - 17.1i)T^{2} \) |
| 41 | \( 1 + (-1.29 - 10.6i)T + (-39.8 + 9.81i)T^{2} \) |
| 43 | \( 1 + (-4.74 + 0.576i)T + (41.7 - 10.2i)T^{2} \) |
| 47 | \( 1 + (-1.31 - 5.34i)T + (-41.6 + 21.8i)T^{2} \) |
| 53 | \( 1 + (-5.01 + 9.55i)T + (-30.1 - 43.6i)T^{2} \) |
| 59 | \( 1 + (2.29 + 2.02i)T + (7.11 + 58.5i)T^{2} \) |
| 61 | \( 1 + (-14.0 - 3.45i)T + (54.0 + 28.3i)T^{2} \) |
| 67 | \( 1 + (-7.04 - 2.67i)T + (50.1 + 44.4i)T^{2} \) |
| 71 | \( 1 + (6.67 + 3.50i)T + (40.3 + 58.4i)T^{2} \) |
| 73 | \( 1 + (-3.04 + 3.43i)T + (-8.79 - 72.4i)T^{2} \) |
| 83 | \( 1 + (-7.76 - 8.76i)T + (-10.0 + 82.3i)T^{2} \) |
| 89 | \( 1 + (9.26 - 4.86i)T + (50.5 - 73.2i)T^{2} \) |
| 97 | \( 1 + (2.45 - 9.96i)T + (-85.8 - 45.0i)T^{2} \) |
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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
−10.40275514117954010542245711397, −9.468430234602717715174830618505, −8.570486420993550780702691323299, −7.979566948326987970770277239746, −6.83753528294820457646207559664, −6.39094280214282449457691845263, −5.29867579785551161945206168973, −3.82150096619501646473661709261, −2.75747023845989666326391573646, −1.18150300910285074192950128141,
0.819008253258840085626500035985, 2.13197149307231242343089077198, 3.95133803459157464843606157184, 4.47925833326030579291098887588, 5.58877055469451106448943042112, 7.20168754262760095207313093220, 7.59583621218150880226633409465, 8.425740644973939639003714274811, 9.404327807635915993790856192415, 10.19016002337927829907447660564
