| L(s) = 1 | − 1.58·3-s + (−1.82 − 1.29i)5-s + i·7-s − 0.475·9-s + 6.23i·11-s − 2.95·13-s + (2.89 + 2.05i)15-s + 5.70i·17-s − 6.34i·19-s − 1.58i·21-s + 2.70i·23-s + (1.64 + 4.72i)25-s + 5.52·27-s − 3.28i·29-s + 8.81·31-s + ⋯ |
| L(s) = 1 | − 0.917·3-s + (−0.814 − 0.579i)5-s + 0.377i·7-s − 0.158·9-s + 1.87i·11-s − 0.819·13-s + (0.747 + 0.531i)15-s + 1.38i·17-s − 1.45i·19-s − 0.346i·21-s + 0.563i·23-s + (0.328 + 0.944i)25-s + 1.06·27-s − 0.610i·29-s + 1.58·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2246603094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2246603094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.82 + 1.29i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.58T + 3T^{2} \) |
| 11 | \( 1 - 6.23iT - 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 5.70iT - 17T^{2} \) |
| 19 | \( 1 + 6.34iT - 19T^{2} \) |
| 23 | \( 1 - 2.70iT - 23T^{2} \) |
| 29 | \( 1 + 3.28iT - 29T^{2} \) |
| 31 | \( 1 - 8.81T + 31T^{2} \) |
| 37 | \( 1 + 0.261T + 37T^{2} \) |
| 41 | \( 1 + 1.95T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 + 8.29iT - 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 0.473iT - 59T^{2} \) |
| 61 | \( 1 - 10.1iT - 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 - 0.374T + 71T^{2} \) |
| 73 | \( 1 + 0.725iT - 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 5.65iT - 97T^{2} \) |
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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
−8.730933719929595194122879154177, −8.029233920452302560617125607278, −7.14952107053303115195526754775, −6.54291442704712635975967030708, −5.46374717350226777820196292779, −4.74990581227602904180388359734, −4.31261446154955726661067717592, −2.88156240776580228051375982185, −1.68068571535984967387876062331, −0.12012690359850711259570557071,
0.855953707800884263043969489795, 2.83987785915542339522757891449, 3.39205990507348175431483722002, 4.58837128141538174622037638150, 5.32034176839021365597684968448, 6.30284380496414300990340704735, 6.69680604446097110421123635445, 7.88628060151788298267585332511, 8.196768669391904117699970123664, 9.300945363772581591982552817942
