| L(s) = 1 | + i·2-s − 4-s − 0.966·5-s − i·8-s − 0.966i·10-s − 2.87i·11-s + 2.13i·13-s + 16-s + 2.44·17-s + 0.528i·19-s + 0.966·20-s + 2.87·22-s + 1.61i·23-s − 4.06·25-s − 2.13·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.432·5-s − 0.353i·8-s − 0.305i·10-s − 0.866i·11-s + 0.591i·13-s + 0.250·16-s + 0.594·17-s + 0.121i·19-s + 0.216·20-s + 0.612·22-s + 0.336i·23-s − 0.813·25-s − 0.418·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6181963131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6181963131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.966T + 5T^{2} \) |
| 11 | \( 1 + 2.87iT - 11T^{2} \) |
| 13 | \( 1 - 2.13iT - 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 0.528iT - 19T^{2} \) |
| 23 | \( 1 - 1.61iT - 23T^{2} \) |
| 29 | \( 1 + 2.43iT - 29T^{2} \) |
| 31 | \( 1 - 0.279iT - 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 + 3.69T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 6.27iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 10.1iT - 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 + 14.2iT - 71T^{2} \) |
| 73 | \( 1 + 1.31iT - 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 - 8.58T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 + 4.27iT - 97T^{2} \) |
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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
−8.450966426333670737582750692982, −7.992005562191243890657803892143, −7.17863577857734975311681205171, −6.41129423015119783461406986707, −5.66947040076510608588267479594, −4.88401902702947733263399569075, −3.87789408702359193576633548238, −3.23782963894280981733760688926, −1.71995657120216965878234440025, −0.21429021688213296418030590924,
1.26219790072297759960609769155, 2.38083560093554622356761926358, 3.35077923922089524257195768806, 4.13342867037622152009896051108, 4.99760605107857833320257869702, 5.75549514555319918993449485367, 6.89557519914851145370121810764, 7.60451304306979137014718309798, 8.336222301876603632427998048019, 9.085338975622148989320931910825
