| L(s) = 1 | − i·3-s − 2i·7-s + 2·9-s − i·13-s − 4i·17-s + 4·19-s − 2·21-s − i·23-s − 5i·27-s + 3·29-s − 31-s − 8i·37-s − 39-s − 5·41-s + 6i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.755i·7-s + 0.666·9-s − 0.277i·13-s − 0.970i·17-s + 0.917·19-s − 0.436·21-s − 0.208i·23-s − 0.962i·27-s + 0.557·29-s − 0.179·31-s − 1.31i·37-s − 0.160·39-s − 0.780·41-s + 0.914i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930800459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930800459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−7.72765576009076395669394797812, −7.47319435499693160416064138720, −6.79340137275650513220145755505, −6.03021444292939359049747667735, −5.05684481265554933910522755670, −4.38317171589450461420408720334, −3.47146228558020806825342370411, −2.55786010747979372161898466282, −1.42222034249926501806183042596, −0.58151288401438805424188346575,
1.27481293674721069818678556489, 2.24731930222627930377552506802, 3.35337560757341363456061478935, 3.98588794714837404924292127733, 4.94079577608740647491928818249, 5.45073658544471289738114492843, 6.39693964969214777244098780027, 7.04356867610614022688941648505, 7.928854697608569505925354632955, 8.650524203935103463082212286254
