| L(s) = 1 | − 5-s + 2.82·7-s − 11-s + 4.82·13-s + 0.828·17-s + 5.65·19-s + 5.65·23-s + 25-s − 2·29-s − 5.65·31-s − 2.82·35-s + 6·37-s + 2·41-s − 1.17·43-s + 5.65·47-s + 1.00·49-s − 0.343·53-s + 55-s + 9.65·59-s + 0.343·61-s − 4.82·65-s − 9.65·67-s − 13.6·71-s + 14.4·73-s − 2.82·77-s − 9.65·79-s + 10.8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.06·7-s − 0.301·11-s + 1.33·13-s + 0.200·17-s + 1.29·19-s + 1.17·23-s + 0.200·25-s − 0.371·29-s − 1.01·31-s − 0.478·35-s + 0.986·37-s + 0.312·41-s − 0.178·43-s + 0.825·47-s + 0.142·49-s − 0.0471·53-s + 0.134·55-s + 1.25·59-s + 0.0439·61-s − 0.598·65-s − 1.17·67-s − 1.62·71-s + 1.69·73-s − 0.322·77-s − 1.08·79-s + 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.549652344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.549652344\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 9.65T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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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
−7.67115755225087040991255137144, −7.45748923837487348170838454977, −6.47965675636674423380211306936, −5.58037631382604466389428790817, −5.14564792742901152204702267269, −4.26090365147742957206211729602, −3.56306200212927683907312870622, −2.76643153465499913079451508680, −1.58747799428073750283565897930, −0.868540129265185569053857668938,
0.868540129265185569053857668938, 1.58747799428073750283565897930, 2.76643153465499913079451508680, 3.56306200212927683907312870622, 4.26090365147742957206211729602, 5.14564792742901152204702267269, 5.58037631382604466389428790817, 6.47965675636674423380211306936, 7.45748923837487348170838454977, 7.67115755225087040991255137144
