| L(s) = 1 | + 0.553·5-s − 1.19·7-s + 0.843·11-s + 2.93·13-s + 3.89·17-s − 3.92·19-s − 3.42·23-s − 4.69·25-s − 8.30·29-s + 7.13·31-s − 0.659·35-s + 5.02·37-s + 8.17·41-s + 9.35·43-s + 6.99·47-s − 5.57·49-s − 4.65·53-s + 0.466·55-s + 5.45·59-s + 1.19·61-s + 1.62·65-s − 5.61·67-s + 10.1·71-s + 3.81·73-s − 1.00·77-s + 2.65·79-s + 7.17·83-s + ⋯ |
| L(s) = 1 | + 0.247·5-s − 0.450·7-s + 0.254·11-s + 0.815·13-s + 0.945·17-s − 0.899·19-s − 0.714·23-s − 0.938·25-s − 1.54·29-s + 1.28·31-s − 0.111·35-s + 0.826·37-s + 1.27·41-s + 1.42·43-s + 1.02·47-s − 0.796·49-s − 0.639·53-s + 0.0629·55-s + 0.710·59-s + 0.152·61-s + 0.201·65-s − 0.686·67-s + 1.20·71-s + 0.446·73-s − 0.114·77-s + 0.298·79-s + 0.787·83-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
| \(L(1)\) |
\(\approx\) |
\(1.975276280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975276280\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.553T + 5T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 - 0.843T + 11T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 + 3.92T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 - 7.13T + 31T^{2} \) |
| 37 | \( 1 - 5.02T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 - 9.35T + 43T^{2} \) |
| 47 | \( 1 - 6.99T + 47T^{2} \) |
| 53 | \( 1 + 4.65T + 53T^{2} \) |
| 59 | \( 1 - 5.45T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 5.61T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 3.81T + 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 - 7.17T + 83T^{2} \) |
| 89 | \( 1 + 0.0346T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
| 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
−7.954606434941981348884198373238, −7.54437006143931630593352086154, −6.39922174306866962311486005739, −6.07942375111473594342393535825, −5.40021446528050034362212898956, −4.19005096334620727745194171770, −3.80864271947729580725778702134, −2.75411487207705185074113563468, −1.87034525859603141157174730118, −0.74435164085782793715320980292,
0.74435164085782793715320980292, 1.87034525859603141157174730118, 2.75411487207705185074113563468, 3.80864271947729580725778702134, 4.19005096334620727745194171770, 5.40021446528050034362212898956, 6.07942375111473594342393535825, 6.39922174306866962311486005739, 7.54437006143931630593352086154, 7.954606434941981348884198373238
