| L(s) = 1 | + 0.870·2-s − 1.24·4-s − 0.709·5-s − 7-s − 2.82·8-s − 0.617·10-s + 3.47·13-s − 0.870·14-s + 0.0306·16-s + 1.34·17-s − 0.503·19-s + 0.882·20-s + 3.69·23-s − 4.49·25-s + 3.02·26-s + 1.24·28-s − 9.21·29-s + 1.39·31-s + 5.67·32-s + 1.17·34-s + 0.709·35-s + 2.24·37-s − 0.437·38-s + 2.00·40-s + 12.4·41-s − 5.60·43-s + 3.21·46-s + ⋯ |
| L(s) = 1 | + 0.615·2-s − 0.621·4-s − 0.317·5-s − 0.377·7-s − 0.997·8-s − 0.195·10-s + 0.962·13-s − 0.232·14-s + 0.00767·16-s + 0.326·17-s − 0.115·19-s + 0.197·20-s + 0.769·23-s − 0.899·25-s + 0.592·26-s + 0.234·28-s − 1.71·29-s + 0.249·31-s + 1.00·32-s + 0.200·34-s + 0.119·35-s + 0.369·37-s − 0.0710·38-s + 0.316·40-s + 1.94·41-s − 0.854·43-s + 0.473·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.870T + 2T^{2} \) |
| 5 | \( 1 + 0.709T + 5T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 1.34T + 17T^{2} \) |
| 19 | \( 1 + 0.503T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 5.60T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 59 | \( 1 + 7.64T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 - 8.53T + 71T^{2} \) |
| 73 | \( 1 - 2.35T + 73T^{2} \) |
| 79 | \( 1 - 4.00T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−7.67483589002209469921766410266, −6.61937147411750694740315678131, −6.02064101280000223637035591449, −5.42000262102512973854631484078, −4.64289282538004663967666274160, −3.77994618523085351439280076600, −3.51339801658782036124493640440, −2.47326748524991650489704820275, −1.17124046046932608003922132603, 0,
1.17124046046932608003922132603, 2.47326748524991650489704820275, 3.51339801658782036124493640440, 3.77994618523085351439280076600, 4.64289282538004663967666274160, 5.42000262102512973854631484078, 6.02064101280000223637035591449, 6.61937147411750694740315678131, 7.67483589002209469921766410266
