| L(s) = 1 | + 2.37·3-s − 3.96·5-s − 7-s + 2.63·9-s − 1.90·11-s − 13-s − 9.41·15-s − 0.790·17-s + 6.58·19-s − 2.37·21-s + 2.28·23-s + 10.7·25-s − 0.858·27-s + 4.83·29-s − 5.20·31-s − 4.51·33-s + 3.96·35-s + 4.52·37-s − 2.37·39-s + 7.73·41-s + 4.64·43-s − 10.4·45-s − 2.38·47-s + 49-s − 1.87·51-s + 10.0·53-s + 7.53·55-s + ⋯ |
| L(s) = 1 | + 1.37·3-s − 1.77·5-s − 0.377·7-s + 0.879·9-s − 0.573·11-s − 0.277·13-s − 2.43·15-s − 0.191·17-s + 1.51·19-s − 0.518·21-s + 0.476·23-s + 2.14·25-s − 0.165·27-s + 0.897·29-s − 0.934·31-s − 0.785·33-s + 0.670·35-s + 0.743·37-s − 0.380·39-s + 1.20·41-s + 0.708·43-s − 1.55·45-s − 0.347·47-s + 0.142·49-s − 0.262·51-s + 1.37·53-s + 1.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.847993878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.847993878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.37T + 3T^{2} \) |
| 5 | \( 1 + 3.96T + 5T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 17 | \( 1 + 0.790T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 + 5.20T + 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 - 7.73T + 41T^{2} \) |
| 43 | \( 1 - 4.64T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 2.77T + 59T^{2} \) |
| 61 | \( 1 - 7.73T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 + 9.41T + 71T^{2} \) |
| 73 | \( 1 + 9.55T + 73T^{2} \) |
| 79 | \( 1 - 7.41T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 9.60T + 97T^{2} \) |
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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
−8.647764395536187730685152120938, −7.968412905419684716665054876779, −7.46509513137120737225751162135, −6.99277813580337793779382992956, −5.57801403968554722200635902166, −4.56070102467209038883266676253, −3.79816651776926371235306715234, −3.14639262885545558542839966205, −2.50221501509316762689421871724, −0.76820676676879764267198453086,
0.76820676676879764267198453086, 2.50221501509316762689421871724, 3.14639262885545558542839966205, 3.79816651776926371235306715234, 4.56070102467209038883266676253, 5.57801403968554722200635902166, 6.99277813580337793779382992956, 7.46509513137120737225751162135, 7.968412905419684716665054876779, 8.647764395536187730685152120938
