| L(s) = 1 | − 3i·7-s + 3·9-s + 6·11-s − 4i·13-s + 5·19-s − 4i·23-s − 2·29-s − 31-s + 2i·37-s − 9·41-s + 2i·43-s − 4i·47-s − 2·49-s + 12i·53-s − 9·59-s + ⋯ |
| L(s) = 1 | − 1.13i·7-s + 9-s + 1.80·11-s − 1.10i·13-s + 1.14·19-s − 0.834i·23-s − 0.371·29-s − 0.179·31-s + 0.328i·37-s − 1.40·41-s + 0.304i·43-s − 0.583i·47-s − 0.285·49-s + 1.64i·53-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.327249455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.327249455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−8.559261498261843471395525988571, −7.65978514813144250419717979225, −7.05990907614649729787142610544, −6.52938690245823920661104046834, −5.50038108381495977191720285530, −4.48136678028281885384523047724, −3.90455622627953533330914549927, −3.12832348544769310182907569357, −1.54121382286091474750138514618, −0.842492000836543627048986398426,
1.35625292098776878499430268414, 1.99911702128363712174137606515, 3.41072079706304510203810180872, 4.04791335013041913763192847543, 5.00029499680304926285784911293, 5.81317112369721490714626764485, 6.74439355084789067170071929296, 7.07422155128492033874022089975, 8.174562538305349882902572345938, 9.061945511880465430432077440277
