| L(s) = 1 | + (−0.5 + 1.65i)3-s + 2·4-s + (1.5 − 1.65i)5-s + (−2.5 − 1.65i)9-s + 3.31i·11-s + (−1 + 3.31i)12-s + (2 + 3.31i)15-s + 4·16-s + (3 − 3.31i)20-s − 9·23-s + (−0.5 − 4.97i)25-s + (4 − 3.31i)27-s − 5·31-s + (−5.5 − 1.65i)33-s + (−5 − 3.31i)36-s − 9.94i·37-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.957i)3-s + 4-s + (0.670 − 0.741i)5-s + (−0.833 − 0.552i)9-s + 1.00i·11-s + (−0.288 + 0.957i)12-s + (0.516 + 0.856i)15-s + 16-s + (0.670 − 0.741i)20-s − 1.87·23-s + (−0.100 − 0.994i)25-s + (0.769 − 0.638i)27-s − 0.898·31-s + (−0.957 − 0.288i)33-s + (−0.833 − 0.552i)36-s − 1.63i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29093 + 0.359111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29093 + 0.359111i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.5 - 1.65i)T \) |
| 5 | \( 1 + (-1.5 + 1.65i)T \) |
| 11 | \( 1 - 3.31iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 9T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 9.94iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 9.94iT - 67T^{2} \) |
| 71 | \( 1 + 16.5iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 - 9.94iT - 97T^{2} \) |
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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
−12.57931310526937732857742718164, −11.96696770341127170360323595282, −10.77062807594740952814869709491, −9.987392923331185778038405413004, −9.127927653632167176705081232588, −7.73025179398020348170243285461, −6.27427733884725445037706944222, −5.37630621914649892911840967897, −4.03353961381990640753076469189, −2.14740718118359859647804121198,
1.85776434273011545659431092329, 3.11665881992685121917386548775, 5.73816012184423975212059417801, 6.30832757896457576293815263842, 7.31904052608451463662153513888, 8.329573024704892966199754961875, 10.00425482064437413577053579651, 10.98894133422688914558602680253, 11.63447741611660777958485544718, 12.67036661082658082052048313460
