| L(s) = 1 | + 2.67·5-s − 3.81·7-s − 3.72·11-s − 3.66·13-s − 0.831·17-s + 5.30·19-s + 7.23·23-s + 2.15·25-s − 4.83·29-s + 8.85·31-s − 10.1·35-s + 6.37·37-s − 9.14·41-s − 3.12·43-s − 0.187·47-s + 7.52·49-s + 1.52·53-s − 9.96·55-s + 2.07·59-s − 14.0·61-s − 9.80·65-s + 7.38·67-s + 0.881·71-s + 3.19·73-s + 14.2·77-s − 1.41·79-s + 0.752·83-s + ⋯ |
| L(s) = 1 | + 1.19·5-s − 1.44·7-s − 1.12·11-s − 1.01·13-s − 0.201·17-s + 1.21·19-s + 1.50·23-s + 0.431·25-s − 0.897·29-s + 1.59·31-s − 1.72·35-s + 1.04·37-s − 1.42·41-s − 0.476·43-s − 0.0273·47-s + 1.07·49-s + 0.209·53-s − 1.34·55-s + 0.269·59-s − 1.80·61-s − 1.21·65-s + 0.902·67-s + 0.104·71-s + 0.373·73-s + 1.61·77-s − 0.158·79-s + 0.0825·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.658878691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658878691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 17 | \( 1 + 0.831T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 - 6.37T + 37T^{2} \) |
| 41 | \( 1 + 9.14T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 0.187T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 2.07T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 - 0.881T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 - 0.752T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 0.441T + 97T^{2} \) |
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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
−7.987074470521276485480032643183, −7.24881992111928534083298291379, −6.63755391798252216796294585389, −5.94508469898266799446230162286, −5.26371801142891544065083640565, −4.70645391902773575411699892317, −3.25299729517588580933011961771, −2.87540093613021672131586759551, −2.02654729665834087933266485529, −0.64903690351395620793815225101,
0.64903690351395620793815225101, 2.02654729665834087933266485529, 2.87540093613021672131586759551, 3.25299729517588580933011961771, 4.70645391902773575411699892317, 5.26371801142891544065083640565, 5.94508469898266799446230162286, 6.63755391798252216796294585389, 7.24881992111928534083298291379, 7.987074470521276485480032643183
