| L(s) = 1 | − 2i·3-s + 2i·7-s − 9-s − 6i·13-s + 2i·17-s + 19-s + 4·21-s + 2i·23-s − 4i·27-s + 2·29-s + 4·31-s − 10i·37-s − 12·39-s − 10·41-s − 6i·43-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + 0.755i·7-s − 0.333·9-s − 1.66i·13-s + 0.485i·17-s + 0.229·19-s + 0.872·21-s + 0.417i·23-s − 0.769i·27-s + 0.371·29-s + 0.718·31-s − 1.64i·37-s − 1.92·39-s − 1.56·41-s − 0.914i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.557522835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557522835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−8.604157004218894647066953018032, −8.215003478037161535612588454734, −7.38085164086044896113199831655, −6.68623316230868009385269762015, −5.73408840322144802552398059789, −5.25878705003116847708138839491, −3.80027802744000710567578750861, −2.75708110867717122872784250534, −1.85608172423703575011480547082, −0.60562547916436570633318595538,
1.35144508543905545735368819591, 2.83337787832829236963395980921, 3.88833863067598626535280703104, 4.49975178305613840369240429382, 5.08425147123881701964406324837, 6.45540485107953424439008618878, 6.95102633656829355135763511278, 8.006229235820168198037045670736, 8.902395607164004577111770790657, 9.589794720351627286948179479459
