| L(s) = 1 | + 3-s − 5-s − 1.63i·7-s + 9-s − 1.02i·11-s + 1.63i·13-s − 15-s + 3.31·17-s + (−4.31 + 0.616i)19-s − 1.63i·21-s + 8.09i·23-s + 25-s + 27-s + 1.02i·29-s + 6.30·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.619i·7-s + 0.333·9-s − 0.308i·11-s + 0.454i·13-s − 0.258·15-s + 0.804·17-s + (−0.989 + 0.141i)19-s − 0.357i·21-s + 1.68i·23-s + 0.200·25-s + 0.192·27-s + 0.189i·29-s + 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.163476353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.163476353\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (4.31 - 0.616i)T \) |
| good | 7 | \( 1 + 1.63iT - 7T^{2} \) |
| 11 | \( 1 + 1.02iT - 11T^{2} \) |
| 13 | \( 1 - 1.63iT - 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 23 | \( 1 - 8.09iT - 23T^{2} \) |
| 29 | \( 1 - 1.02iT - 29T^{2} \) |
| 31 | \( 1 - 6.30T + 31T^{2} \) |
| 37 | \( 1 + 9.11iT - 37T^{2} \) |
| 41 | \( 1 - 2.25iT - 41T^{2} \) |
| 43 | \( 1 - 9.11iT - 43T^{2} \) |
| 47 | \( 1 + 6.85iT - 47T^{2} \) |
| 53 | \( 1 - 8.09iT - 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 + 0.989T + 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 5.22iT - 89T^{2} \) |
| 97 | \( 1 - 19.0iT - 97T^{2} \) |
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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
−8.208170396800516097423878586521, −7.70530662340951677115171129139, −7.07092128219278479003935091933, −6.25255305379225626826325194339, −5.37507971820310973793835195149, −4.37888076703286065574120298561, −3.79272912613498347557269915276, −3.07101289912417136547998943364, −1.95370254406513506843082271591, −0.877639175877873802871641110875,
0.73503334738773033826815763022, 2.15085587367478874363690125709, 2.79212808414578053496172012019, 3.71965991057708733207978738664, 4.54026111633040549717034390398, 5.24644062281589103412411007628, 6.29151700791426540118001018363, 6.82680084129617409800513168951, 7.81013389694414752628279523442, 8.365475185158954573640701078064
