| L(s) = 1 | + 1.08·5-s + (2.10 − 1.60i)7-s + 1.23i·11-s − 3.69i·13-s + 6.30·17-s − 7.76i·19-s − 7.17i·23-s − 3.82·25-s + 1.41i·29-s + 4.54i·31-s + (2.27 − 1.74i)35-s − 2.82·37-s − 9.37·41-s − 7.68·43-s + 10.9·47-s + ⋯ |
| L(s) = 1 | + 0.484·5-s + (0.794 − 0.607i)7-s + 0.371i·11-s − 1.02i·13-s + 1.53·17-s − 1.78i·19-s − 1.49i·23-s − 0.765·25-s + 0.262i·29-s + 0.816i·31-s + (0.384 − 0.294i)35-s − 0.464·37-s − 1.46·41-s − 1.17·43-s + 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0377 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0377 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.123611465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.123611465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.10 + 1.60i)T \) |
| good | 5 | \( 1 - 1.08T + 5T^{2} \) |
| 11 | \( 1 - 1.23iT - 11T^{2} \) |
| 13 | \( 1 + 3.69iT - 13T^{2} \) |
| 17 | \( 1 - 6.30T + 17T^{2} \) |
| 19 | \( 1 + 7.76iT - 19T^{2} \) |
| 23 | \( 1 + 7.17iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 4.54iT - 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 9.37T + 41T^{2} \) |
| 43 | \( 1 + 7.68T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 6.43T + 59T^{2} \) |
| 61 | \( 1 - 9.55iT - 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 10.6iT - 71T^{2} \) |
| 73 | \( 1 - 4.59iT - 73T^{2} \) |
| 79 | \( 1 - 9.42T + 79T^{2} \) |
| 83 | \( 1 + 1.88T + 83T^{2} \) |
| 89 | \( 1 - 5.04T + 89T^{2} \) |
| 97 | \( 1 - 5.86iT - 97T^{2} \) |
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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
−8.284629282219602926092856607281, −7.47725557442644655671373564149, −6.94939487070963578394397713385, −5.99800951014246258988137922914, −5.10669511908576180517657147633, −4.73529153544703388263614516864, −3.56478368361653678779367538222, −2.72400841823022919480616967822, −1.64721929753368304515221842897, −0.60182253749267206975487565159,
1.48261040184393854282318609858, 1.91441282110078396810034592695, 3.26156962932312289030772627705, 3.96119020732417917935333086599, 5.05338379137258549594092935849, 5.76730093946242735900294401421, 6.09016181019842226447534669082, 7.36861965065333639207105094971, 7.87974563862019870518966753819, 8.557952639535826529324107795712
