| L(s) = 1 | + 3-s − i·5-s − 4.60i·7-s + 9-s + 3.60·13-s − i·15-s + 4.60·17-s − 4.60i·19-s − 4.60i·21-s + 1.39·23-s − 25-s + 27-s + 4.60·29-s + 6i·31-s − 4.60·35-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447i·5-s − 1.74i·7-s + 0.333·9-s + 1.00·13-s − 0.258i·15-s + 1.11·17-s − 1.05i·19-s − 1.00i·21-s + 0.290·23-s − 0.200·25-s + 0.192·27-s + 0.855·29-s + 1.07i·31-s − 0.778·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.474839296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.474839296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 - 3.60T \) |
| good | 7 | \( 1 + 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 + 4.60iT - 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 - 9.21iT - 37T^{2} \) |
| 41 | \( 1 + 3.21iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 9.21iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9.21iT - 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 3.21iT - 67T^{2} \) |
| 71 | \( 1 - 9.21iT - 71T^{2} \) |
| 73 | \( 1 - 1.39iT - 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 2.78iT - 83T^{2} \) |
| 89 | \( 1 + 15.2iT - 89T^{2} \) |
| 97 | \( 1 + 1.39iT - 97T^{2} \) |
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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
−8.404967451470326921935720208736, −7.88558636825873937107037012582, −6.97249968597516623823760828729, −6.56336110816124090993235560480, −5.22432300630744085807271486023, −4.55554393744719937886631465516, −3.65829968054916541797878799124, −3.11129097390907533998732608433, −1.52465282390988469902065248244, −0.77934816697333602416929628104,
1.44306699330090206229331145769, 2.45302537897233424884970417956, 3.16671138678413578761732527471, 3.98132628534948689142885199852, 5.21417508815384587076344547370, 5.93478562722267501611468235224, 6.40974405480233207025664721891, 7.68824956411235090947198019191, 8.098005311842368412274551149403, 8.921635644161292194855927223226
