| L(s) = 1 | + 2-s − 4-s − 1.41·5-s − 3·8-s − 1.41·10-s − 2.82·11-s + 2·13-s − 16-s + 7.07·17-s + 2.82·19-s + 1.41·20-s − 2.82·22-s − 2.99·25-s + 2·26-s + 2·29-s − 4·31-s + 5·32-s + 7.07·34-s + 4.24·37-s + 2.82·38-s + 4.24·40-s + 8·41-s − 2.82·43-s + 2.82·44-s − 4·47-s − 7·49-s − 2.99·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.5·4-s − 0.632·5-s − 1.06·8-s − 0.447·10-s − 0.852·11-s + 0.554·13-s − 0.250·16-s + 1.71·17-s + 0.648·19-s + 0.316·20-s − 0.603·22-s − 0.599·25-s + 0.392·26-s + 0.371·29-s − 0.718·31-s + 0.883·32-s + 1.21·34-s + 0.697·37-s + 0.458·38-s + 0.670·40-s + 1.24·41-s − 0.431·43-s + 0.426·44-s − 0.583·47-s − 49-s − 0.424·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 9.89T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−7.75884186562632037740789552429, −7.48566354498520225951484202579, −6.10168237150711832147909351027, −5.70902816763091833724496468900, −4.91119587874150894284577538337, −4.19467484172870547336532659780, −3.37266891241874883591428904301, −2.88418907881095327022984268099, −1.28281217120786705521251084897, 0,
1.28281217120786705521251084897, 2.88418907881095327022984268099, 3.37266891241874883591428904301, 4.19467484172870547336532659780, 4.91119587874150894284577538337, 5.70902816763091833724496468900, 6.10168237150711832147909351027, 7.48566354498520225951484202579, 7.75884186562632037740789552429
