| L(s) = 1 | − 5-s − 5.18·7-s − 5.53·11-s + 1.78·13-s + 2.22·17-s − 8.22·19-s − 23-s + 25-s − 7.06·29-s − 2.06·31-s + 5.18·35-s − 5.02·37-s − 11.4·41-s − 8.62·43-s − 1.37·47-s + 19.9·49-s + 4.46·53-s + 5.53·55-s − 11.1·59-s + 7.81·61-s − 1.78·65-s − 1.96·67-s + 8.51·71-s − 10.7·73-s + 28.6·77-s − 14.0·79-s − 6.83·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.96·7-s − 1.66·11-s + 0.496·13-s + 0.539·17-s − 1.88·19-s − 0.208·23-s + 0.200·25-s − 1.31·29-s − 0.371·31-s + 0.876·35-s − 0.826·37-s − 1.79·41-s − 1.31·43-s − 0.200·47-s + 2.84·49-s + 0.613·53-s + 0.745·55-s − 1.45·59-s + 1.00·61-s − 0.221·65-s − 0.240·67-s + 1.01·71-s − 1.25·73-s + 3.26·77-s − 1.57·79-s − 0.750·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05780330460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05780330460\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 5.18T + 7T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 19 | \( 1 + 8.22T + 19T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8.62T + 43T^{2} \) |
| 47 | \( 1 + 1.37T + 47T^{2} \) |
| 53 | \( 1 - 4.46T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.81T + 61T^{2} \) |
| 67 | \( 1 + 1.96T + 67T^{2} \) |
| 71 | \( 1 - 8.51T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 4.19T + 97T^{2} \) |
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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
−7.79718138311057409096675919620, −7.02144087933824976575810742517, −6.50106794601717640382071702906, −5.77418269124391140245653410707, −5.14795285629342753355388132269, −4.05597544588019902193498185721, −3.44480098219122757321893823108, −2.84913517630237701829382019422, −1.89312989709893550645436675065, −0.10897282743110876172984305429,
0.10897282743110876172984305429, 1.89312989709893550645436675065, 2.84913517630237701829382019422, 3.44480098219122757321893823108, 4.05597544588019902193498185721, 5.14795285629342753355388132269, 5.77418269124391140245653410707, 6.50106794601717640382071702906, 7.02144087933824976575810742517, 7.79718138311057409096675919620
