| L(s) = 1 | + i·7-s + 6·11-s − 2i·13-s + 4·19-s − 6i·23-s + 6·29-s + 8·31-s + 2i·37-s − 12·41-s + 4i·43-s − 12i·47-s − 49-s − 6i·53-s − 10·61-s + 8i·67-s + ⋯ |
| L(s) = 1 | + 0.377i·7-s + 1.80·11-s − 0.554i·13-s + 0.917·19-s − 1.25i·23-s + 1.11·29-s + 1.43·31-s + 0.328i·37-s − 1.87·41-s + 0.609i·43-s − 1.75i·47-s − 0.142·49-s − 0.824i·53-s − 1.28·61-s + 0.977i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.399372177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.399372177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.248811662233940481852217313048, −7.09202293056876309228527824723, −6.58274237140609938530377426043, −6.01516525757656425775262953203, −5.03256300300425476626843118864, −4.42735300891593009444764014884, −3.48434347903952138590673767717, −2.81514690297851522087089700088, −1.66048934383948113939919306771, −0.74438357632839803568015666113,
1.03261491511701919146318739714, 1.65603105868881597132470199974, 2.99850710010784147513581177145, 3.69508927659995564259377443052, 4.43040979552528500862834811718, 5.12358101024344324129832713009, 6.27549974579458084361128064107, 6.52017477764552950444850539455, 7.38827832652121158513470294345, 7.990084037998074328060509500241
