| L(s) = 1 | − 5-s + 3·7-s + 3·11-s − 13-s − 3·17-s + 4·19-s + 3·23-s + 25-s + 10·29-s + 6·31-s − 3·35-s + 5·37-s + 5·41-s + 2·43-s − 2·47-s + 2·49-s + 11·53-s − 3·55-s + 4·59-s − 61-s + 65-s − 4·67-s − 3·71-s + 6·73-s + 9·77-s − 3·79-s + 16·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.904·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s + 0.625·23-s + 1/5·25-s + 1.85·29-s + 1.07·31-s − 0.507·35-s + 0.821·37-s + 0.780·41-s + 0.304·43-s − 0.291·47-s + 2/7·49-s + 1.51·53-s − 0.404·55-s + 0.520·59-s − 0.128·61-s + 0.124·65-s − 0.488·67-s − 0.356·71-s + 0.702·73-s + 1.02·77-s − 0.337·79-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.368202539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.368202539\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.97290514236963, −14.23246593272111, −14.01448864094597, −13.40242219557712, −12.70608901824452, −12.06166559671977, −11.66920700545516, −11.38628419757407, −10.69355861616082, −10.18501545754530, −9.466006799825704, −8.930630737380632, −8.414961384881553, −7.899600082948979, −7.347067716367268, −6.717673595392525, −6.229547761202106, −5.379539239721063, −4.697495752223130, −4.452780188213476, −3.692176749919461, −2.843829479972202, −2.262360014926309, −1.213568556601413, −0.7999837175867387,
0.7999837175867387, 1.213568556601413, 2.262360014926309, 2.843829479972202, 3.692176749919461, 4.452780188213476, 4.697495752223130, 5.379539239721063, 6.229547761202106, 6.717673595392525, 7.347067716367268, 7.899600082948979, 8.414961384881553, 8.930630737380632, 9.466006799825704, 10.18501545754530, 10.69355861616082, 11.38628419757407, 11.66920700545516, 12.06166559671977, 12.70608901824452, 13.40242219557712, 14.01448864094597, 14.23246593272111, 14.97290514236963
