| L(s) = 1 | − 1.33·3-s + 2.63·5-s − 1.22·9-s + 3.39·11-s + 2.73·13-s − 3.51·15-s − 17-s − 0.404·19-s + 1.46·23-s + 1.94·25-s + 5.62·27-s + 5.14·29-s − 5.78·31-s − 4.52·33-s + 3.35·37-s − 3.64·39-s + 8.88·41-s − 7.39·43-s − 3.22·45-s + 10.8·47-s + 1.33·51-s − 5.85·53-s + 8.95·55-s + 0.538·57-s − 2.15·59-s + 1.64·61-s + 7.20·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 1.17·5-s − 0.407·9-s + 1.02·11-s + 0.758·13-s − 0.906·15-s − 0.242·17-s − 0.0927·19-s + 0.306·23-s + 0.388·25-s + 1.08·27-s + 0.955·29-s − 1.03·31-s − 0.788·33-s + 0.551·37-s − 0.583·39-s + 1.38·41-s − 1.12·43-s − 0.480·45-s + 1.58·47-s + 0.186·51-s − 0.804·53-s + 1.20·55-s + 0.0713·57-s − 0.280·59-s + 0.211·61-s + 0.894·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877441092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877441092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 19 | \( 1 + 0.404T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 + 7.39T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.85T + 53T^{2} \) |
| 59 | \( 1 + 2.15T + 59T^{2} \) |
| 61 | \( 1 - 1.64T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 5.10T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 1.64T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 5.01T + 89T^{2} \) |
| 97 | \( 1 + 1.08T + 97T^{2} \) |
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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
−8.988038778088120228695426020352, −7.892165173707304293127830946335, −6.80104675350490893402772286688, −6.21820582768612079921801213872, −5.80897560335628165379811136194, −4.98123645088500077148172891073, −4.05140204787000637561262632148, −2.96466516312116932419056124281, −1.88753429925440673377817815430, −0.885259736099810619523120277286,
0.885259736099810619523120277286, 1.88753429925440673377817815430, 2.96466516312116932419056124281, 4.05140204787000637561262632148, 4.98123645088500077148172891073, 5.80897560335628165379811136194, 6.21820582768612079921801213872, 6.80104675350490893402772286688, 7.892165173707304293127830946335, 8.988038778088120228695426020352
