| L(s) = 1 | + (2.17 + 0.539i)5-s − i·7-s − 5.41·11-s + 4.34i·13-s + 1.07i·17-s − 4.34·19-s − 6.34i·23-s + (4.41 + 2.34i)25-s − 8.83·29-s − 4.34·31-s + (0.539 − 2.17i)35-s + 8.68i·37-s − 8.34·41-s + 6.15i·43-s − 6.83i·47-s + ⋯ |
| L(s) = 1 | + (0.970 + 0.241i)5-s − 0.377i·7-s − 1.63·11-s + 1.20i·13-s + 0.261i·17-s − 0.995·19-s − 1.32i·23-s + (0.883 + 0.468i)25-s − 1.64·29-s − 0.779·31-s + (0.0911 − 0.366i)35-s + 1.42i·37-s − 1.30·41-s + 0.938i·43-s − 0.997i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3564419069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3564419069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.17 - 0.539i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 4.34iT - 13T^{2} \) |
| 17 | \( 1 - 1.07iT - 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 6.34iT - 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 - 8.68iT - 37T^{2} \) |
| 41 | \( 1 + 8.34T + 41T^{2} \) |
| 43 | \( 1 - 6.15iT - 43T^{2} \) |
| 47 | \( 1 + 6.83iT - 47T^{2} \) |
| 53 | \( 1 + 6.18iT - 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 11.1iT - 73T^{2} \) |
| 79 | \( 1 + 0.680T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
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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
−9.320242712289931058559898333498, −8.509826673041543902938927368016, −7.77328341163687838646927342708, −6.77285867514921138156500711782, −6.33986870572924351825375377067, −5.28931250826951267184726123926, −4.68581386536221207652273383825, −3.55592530133416368806222763975, −2.42363650973790615175832615393, −1.77049294945853635274867955725,
0.10272849062823947252017543617, 1.78946046660152835524318613624, 2.58011024298712193314680025982, 3.52452300524485334027501002105, 4.87787988195624605664006181936, 5.63948463757538362062992275367, 5.78608212811687808460838916957, 7.17717200424019741746383155146, 7.78384615566134339167288637722, 8.602884350599730264582457257345
