| L(s) = 1 | − 3-s + 1.94·5-s + 3.19·7-s + 9-s + 0.785·11-s − 4.88·13-s − 1.94·15-s + 2.74·17-s − 1.67·19-s − 3.19·21-s + 6.23·23-s − 1.21·25-s − 27-s − 0.902·29-s − 5.13·31-s − 0.785·33-s + 6.21·35-s + 0.279·37-s + 4.88·39-s + 1.15·41-s − 4.50·43-s + 1.94·45-s + 8.29·47-s + 3.22·49-s − 2.74·51-s − 6.24·53-s + 1.52·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.869·5-s + 1.20·7-s + 0.333·9-s + 0.236·11-s − 1.35·13-s − 0.502·15-s + 0.666·17-s − 0.384·19-s − 0.697·21-s + 1.30·23-s − 0.243·25-s − 0.192·27-s − 0.167·29-s − 0.922·31-s − 0.136·33-s + 1.05·35-s + 0.0459·37-s + 0.782·39-s + 0.180·41-s − 0.687·43-s + 0.289·45-s + 1.21·47-s + 0.460·49-s − 0.384·51-s − 0.858·53-s + 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.299811632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299811632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.94T + 5T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 - 0.785T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 - 2.74T + 17T^{2} \) |
| 19 | \( 1 + 1.67T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 + 0.902T + 29T^{2} \) |
| 31 | \( 1 + 5.13T + 31T^{2} \) |
| 37 | \( 1 - 0.279T + 37T^{2} \) |
| 41 | \( 1 - 1.15T + 41T^{2} \) |
| 43 | \( 1 + 4.50T + 43T^{2} \) |
| 47 | \( 1 - 8.29T + 47T^{2} \) |
| 53 | \( 1 + 6.24T + 53T^{2} \) |
| 59 | \( 1 - 9.18T + 59T^{2} \) |
| 61 | \( 1 - 0.717T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 2.50T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 1.53T + 97T^{2} \) |
| 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
−7.75077282465788190439582106846, −7.11201565444419446678736040955, −6.47015239431818098749578828188, −5.45886358573859670096875546339, −5.23688935479547136116978659706, −4.55564645297523329743496884835, −3.57903204346494553360185194012, −2.37716355209986162408921028392, −1.81068800750492243838849535684, −0.792429339941357491478249837240,
0.792429339941357491478249837240, 1.81068800750492243838849535684, 2.37716355209986162408921028392, 3.57903204346494553360185194012, 4.55564645297523329743496884835, 5.23688935479547136116978659706, 5.45886358573859670096875546339, 6.47015239431818098749578828188, 7.11201565444419446678736040955, 7.75077282465788190439582106846
