| L(s) = 1 | + 2.23i·3-s − 4.47i·7-s − 2.00·9-s − 2.23·11-s − 4i·13-s − 7i·17-s − 6.70·19-s + 10.0·21-s − 4.47i·23-s + 2.23i·27-s − 4.47·31-s − 5.00i·33-s + 2i·37-s + 8.94·39-s + 5·41-s + ⋯ |
| L(s) = 1 | + 1.29i·3-s − 1.69i·7-s − 0.666·9-s − 0.674·11-s − 1.10i·13-s − 1.69i·17-s − 1.53·19-s + 2.18·21-s − 0.932i·23-s + 0.430i·27-s − 0.803·31-s − 0.870i·33-s + 0.328i·37-s + 1.43·39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.891694 - 0.551097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.891694 - 0.551097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.23iT - 3T^{2} \) |
| 7 | \( 1 + 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8.94iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 2.23iT - 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 9iT - 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−10.22126893401987219877113284521, −9.584141881149224723783467051629, −8.469176036014149012714456319255, −7.56745687713949154328048309048, −6.74252003028005576873378520665, −5.34854868382741696583165440957, −4.57272530213011923377055113525, −3.85967026668060147693194313413, −2.75877629512651577296731893493, −0.50057434318881295746733251681,
1.85793490824930887371118267577, 2.30465757592091406978958476340, 3.95401355288388500054661419207, 5.41618953321174318252451866652, 6.13917781042818966015678213445, 6.84081540251077787635520578612, 7.971796019656869401796858542685, 8.552610306597801414122356193401, 9.305065441663806047337678323255, 10.53520457198126335357522580774
