| L(s) = 1 | + 1.32i·3-s + i·5-s + 3.37·7-s + 1.23·9-s − 1.86·11-s − 4.25·13-s − 1.32·15-s + 3.38i·17-s − 4.52·19-s + 4.48i·21-s + (−3.47 + 3.30i)23-s − 25-s + 5.62i·27-s + 3.15·29-s + 2.04i·31-s + ⋯ |
| L(s) = 1 | + 0.767i·3-s + 0.447i·5-s + 1.27·7-s + 0.411·9-s − 0.563·11-s − 1.17·13-s − 0.343·15-s + 0.821i·17-s − 1.03·19-s + 0.978i·21-s + (−0.725 + 0.688i)23-s − 0.200·25-s + 1.08i·27-s + 0.585·29-s + 0.367i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.527101935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527101935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (3.47 - 3.30i)T \) |
| good | 3 | \( 1 - 1.32iT - 3T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 - 3.38iT - 17T^{2} \) |
| 19 | \( 1 + 4.52T + 19T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 31 | \( 1 - 2.04iT - 31T^{2} \) |
| 37 | \( 1 - 5.40iT - 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.07T + 43T^{2} \) |
| 47 | \( 1 - 5.06iT - 47T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 - 11.1iT - 59T^{2} \) |
| 61 | \( 1 - 4.01iT - 61T^{2} \) |
| 67 | \( 1 + 5.68T + 67T^{2} \) |
| 71 | \( 1 + 1.69iT - 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 + 4.52T + 79T^{2} \) |
| 83 | \( 1 + 3.81T + 83T^{2} \) |
| 89 | \( 1 - 4.32iT - 89T^{2} \) |
| 97 | \( 1 - 8.95iT - 97T^{2} \) |
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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
−9.755351957297234761130689849919, −8.723313488555222628849605025051, −7.941405549389160106335827244419, −7.36881928207128248411487145721, −6.31750667024767121462849850588, −5.27002444209269445963606259251, −4.60577959597848484410353431732, −3.93829220403334012190739579378, −2.64015025101401934151004511054, −1.64399116401946051274490420310,
0.54419348073239857688382607005, 1.88578855933244745744837535865, 2.51327026008200019510191530816, 4.31666776982475842206503457415, 4.74023121740060723237378222153, 5.69919770439785161773994688405, 6.72866870238882428880224313382, 7.60682717701629008187221712557, 7.928383859009925053781337920537, 8.792783755764358558783643905523
