| L(s) = 1 | − 1.19·3-s − 1.56·5-s + 3.07·7-s − 1.56·9-s − 5.47·11-s + 1.87·15-s − 2.12·17-s − 7.34·19-s − 3.68·21-s − 3.59·23-s − 2.56·25-s + 5.47·27-s − 5·29-s + 6.67·31-s + 6.56·33-s − 4.79·35-s + 6.12·37-s − 4.12·41-s − 0.673·43-s + 2.43·45-s − 0.525·47-s + 2.43·49-s + 2.54·51-s + 1.56·53-s + 8.54·55-s + 8.80·57-s − 10.2·59-s + ⋯ |
| L(s) = 1 | − 0.692·3-s − 0.698·5-s + 1.16·7-s − 0.520·9-s − 1.64·11-s + 0.483·15-s − 0.514·17-s − 1.68·19-s − 0.804·21-s − 0.750·23-s − 0.512·25-s + 1.05·27-s − 0.928·29-s + 1.19·31-s + 1.14·33-s − 0.810·35-s + 1.00·37-s − 0.643·41-s − 0.102·43-s + 0.363·45-s − 0.0767·47-s + 0.348·49-s + 0.356·51-s + 0.214·53-s + 1.15·55-s + 1.16·57-s − 1.33·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5438377099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5438377099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 - 6.12T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 0.673T + 43T^{2} \) |
| 47 | \( 1 + 0.525T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 - 7.34T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 8.54T + 83T^{2} \) |
| 89 | \( 1 - 1.68T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.111373452089270108178652994465, −7.74400365637831444202435678495, −6.64543373711675980821526381150, −5.96248438300879152999021046802, −5.18006601446849506787489692288, −4.65426028802467556753714853989, −3.91595699164119802455057400621, −2.66543677917204874676476810142, −1.96351248886836598003606993016, −0.38803538483078496095530353821,
0.38803538483078496095530353821, 1.96351248886836598003606993016, 2.66543677917204874676476810142, 3.91595699164119802455057400621, 4.65426028802467556753714853989, 5.18006601446849506787489692288, 5.96248438300879152999021046802, 6.64543373711675980821526381150, 7.74400365637831444202435678495, 8.111373452089270108178652994465
