| L(s) = 1 | − 5-s + 1.41·7-s + 11-s − 0.585·13-s + 3.41·17-s − 5.65·19-s + 2.82·23-s + 25-s − 0.828·29-s + 6.48·31-s − 1.41·35-s − 7.65·37-s + 10.4·41-s + 2.58·43-s − 2.82·47-s − 5·49-s + 7.17·53-s − 55-s + 10.4·59-s − 3.17·61-s + 0.585·65-s − 8.48·67-s + 3.17·71-s + 7.41·73-s + 1.41·77-s + 1.65·79-s − 0.242·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.534·7-s + 0.301·11-s − 0.162·13-s + 0.828·17-s − 1.29·19-s + 0.589·23-s + 0.200·25-s − 0.153·29-s + 1.16·31-s − 0.239·35-s − 1.25·37-s + 1.63·41-s + 0.394·43-s − 0.412·47-s − 0.714·49-s + 0.985·53-s − 0.134·55-s + 1.36·59-s − 0.406·61-s + 0.0726·65-s − 1.03·67-s + 0.376·71-s + 0.867·73-s + 0.161·77-s + 0.186·79-s − 0.0266·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.832119334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.832119334\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.41T + 7T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 7.17T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 0.242T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 - 0.828T + 97T^{2} \) |
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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
−8.410506837921309670529534113244, −7.79745959588708426682385365023, −7.04327952103669924828822244729, −6.30481694063817133506824655184, −5.42401850582041045434216929504, −4.61725189832187944956222852994, −3.94621828228751804059354460623, −2.98562958700781800534273445151, −1.95827998990836041620355231472, −0.790819635221559233414311353560,
0.790819635221559233414311353560, 1.95827998990836041620355231472, 2.98562958700781800534273445151, 3.94621828228751804059354460623, 4.61725189832187944956222852994, 5.42401850582041045434216929504, 6.30481694063817133506824655184, 7.04327952103669924828822244729, 7.79745959588708426682385365023, 8.410506837921309670529534113244
