| L(s) = 1 | + 2.23i·5-s − 5.16·7-s − 3.05i·11-s − 0.837·13-s + 6.32·19-s − 4.47i·23-s − 5.00·25-s − 11.5i·35-s + 11.1·37-s − 10.3i·41-s − 2.82i·47-s + 19.6·49-s − 5.65i·53-s + 6.83·55-s + 5.42i·59-s + ⋯ |
| L(s) = 1 | + 0.999i·5-s − 1.95·7-s − 0.921i·11-s − 0.232·13-s + 1.45·19-s − 0.932i·23-s − 1.00·25-s − 1.95i·35-s + 1.83·37-s − 1.61i·41-s − 0.412i·47-s + 2.80·49-s − 0.777i·53-s + 0.921·55-s + 0.706i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9688576324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9688576324\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 5.16T + 7T^{2} \) |
| 11 | \( 1 + 3.05iT - 11T^{2} \) |
| 13 | \( 1 + 0.837T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 5.42iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 18.8iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.587566037581212684215541814342, −8.753548249334104677676748820183, −7.58214323380098597009333803515, −6.89845679903680299981761959023, −6.19593111078127840687493827064, −5.57279802418148079667172963629, −3.97123820910979095905681199479, −3.18456104349816428112560174117, −2.60541316654075141986822361707, −0.45560554574517857371190998410,
1.07979811851267852312989976256, 2.65762034379853982747210459848, 3.61014347285320100181307527789, 4.59253946468535292731181360001, 5.56576427527987491155348479911, 6.33199654303336368596518891403, 7.26622776984638837963754006611, 7.955303149097404691666487797000, 9.249869414804400449985690873880, 9.586429271556884924349139236610
