L(s) = 1 | + (0.593 − 1.02i)2-s + (0.298 + 0.172i)3-s + (0.295 + 0.511i)4-s + (1.71 + 1.44i)5-s + (0.354 − 0.204i)6-s + (−1.01 − 1.75i)7-s + 3.07·8-s + (−1.44 − 2.49i)9-s + (2.49 − 0.903i)10-s + (3.36 + 1.94i)11-s + 0.203i·12-s − 2.40·14-s + (0.262 + 0.725i)15-s + (1.23 − 2.14i)16-s + (4.71 − 2.72i)17-s − 3.42·18-s + ⋯ |
L(s) = 1 | + (0.419 − 0.727i)2-s + (0.172 + 0.0996i)3-s + (0.147 + 0.255i)4-s + (0.764 + 0.644i)5-s + (0.144 − 0.0836i)6-s + (−0.383 − 0.664i)7-s + 1.08·8-s + (−0.480 − 0.831i)9-s + (0.789 − 0.285i)10-s + (1.01 + 0.585i)11-s + 0.0588i·12-s − 0.644·14-s + (0.0678 + 0.187i)15-s + (0.308 − 0.535i)16-s + (1.14 − 0.660i)17-s − 0.806·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.54392 - 0.574983i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.54392 - 0.574983i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.71 - 1.44i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.593 + 1.02i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.298 - 0.172i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.01 + 1.75i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.36 - 1.94i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.71 + 2.72i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.09 - 2.94i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.298 - 0.172i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.18iT - 31T^{2} \) |
| 37 | \( 1 + (2.72 - 4.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.156 + 0.0902i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.15 - 0.669i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 2.42iT - 53T^{2} \) |
| 59 | \( 1 + (6.11 - 3.53i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.38 + 5.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.20 - 3.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.62 - 0.940i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 8.86T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.83T + 83T^{2} \) |
| 89 | \( 1 + (10.6 + 6.12i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.90 + 5.02i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.14774816226722000876123937585, −9.657843758715776025019246648414, −8.578833411279563103351741005069, −7.34635488608194484192352737921, −6.72113345317583296694080139780, −5.82180367772448758786191996379, −4.33808652290219612321678511206, −3.54960932080737533943897638877, −2.74558047071514686826158406442, −1.46102601268888514709824723051,
1.43174174242476147345178175188, 2.58008311296095138606583941437, 4.16995224303630024037355393566, 5.27602501953415845581760165299, 5.87260238037536807891292763809, 6.47452755937809096583636381798, 7.62684270352001406567860293908, 8.656581517687279601035814022649, 9.128762381871197933224555289690, 10.30925779316744060103358953215