| L(s) = 1 | − 0.456·5-s + 1.79·7-s + 0.456·11-s − 13-s − 1.73·17-s − 4.58·19-s + 1.27·23-s − 4.79·25-s + 5.74·29-s − 2.79·31-s − 0.818·35-s − 3.79·37-s + 4.83·41-s − 6.58·43-s − 6.10·47-s − 3.79·49-s − 9.66·53-s − 0.208·55-s + 9.57·59-s + 3.37·61-s + 0.456·65-s + 0.373·67-s + 3.55·71-s − 8.58·73-s + 0.818·77-s − 79-s − 14.8·83-s + ⋯ |
| L(s) = 1 | − 0.204·5-s + 0.677·7-s + 0.137·11-s − 0.277·13-s − 0.420·17-s − 1.05·19-s + 0.265·23-s − 0.958·25-s + 1.06·29-s − 0.501·31-s − 0.138·35-s − 0.623·37-s + 0.755·41-s − 1.00·43-s − 0.891·47-s − 0.541·49-s − 1.32·53-s − 0.0281·55-s + 1.24·59-s + 0.431·61-s + 0.0566·65-s + 0.0456·67-s + 0.422·71-s − 1.00·73-s + 0.0932·77-s − 0.112·79-s − 1.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 0.456T + 5T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 11 | \( 1 - 0.456T + 11T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 - 5.74T + 29T^{2} \) |
| 31 | \( 1 + 2.79T + 31T^{2} \) |
| 37 | \( 1 + 3.79T + 37T^{2} \) |
| 41 | \( 1 - 4.83T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 + 6.10T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 - 0.373T + 67T^{2} \) |
| 71 | \( 1 - 3.55T + 71T^{2} \) |
| 73 | \( 1 + 8.58T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−8.261698004295773224856908665515, −7.28861256520750028812002847962, −6.64150293449265513512957437575, −5.83921749165028698432281984336, −4.90404979665939861942129939601, −4.35916202819393955866705197078, −3.43852064888845700533580840644, −2.36242686584045430205387364304, −1.49892284996558057299343560405, 0,
1.49892284996558057299343560405, 2.36242686584045430205387364304, 3.43852064888845700533580840644, 4.35916202819393955866705197078, 4.90404979665939861942129939601, 5.83921749165028698432281984336, 6.64150293449265513512957437575, 7.28861256520750028812002847962, 8.261698004295773224856908665515
