L(s) = 1 | + 2-s − 3-s + 4-s − 0.801·5-s − 6-s − 7-s + 8-s + 9-s − 0.801·10-s + 4.04·11-s − 12-s − 14-s + 0.801·15-s + 16-s − 0.753·17-s + 18-s + 0.664·19-s − 0.801·20-s + 21-s + 4.04·22-s − 3.80·23-s − 24-s − 4.35·25-s − 27-s − 28-s − 8.74·29-s + 0.801·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.358·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.253·10-s + 1.22·11-s − 0.288·12-s − 0.267·14-s + 0.207·15-s + 0.250·16-s − 0.182·17-s + 0.235·18-s + 0.152·19-s − 0.179·20-s + 0.218·21-s + 0.863·22-s − 0.792·23-s − 0.204·24-s − 0.871·25-s − 0.192·27-s − 0.188·28-s − 1.62·29-s + 0.146·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 0.801T + 5T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 17 | \( 1 + 0.753T + 17T^{2} \) |
| 19 | \( 1 - 0.664T + 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 0.335T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 - 6.89T + 41T^{2} \) |
| 43 | \( 1 + 1.30T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 - 3.04T + 53T^{2} \) |
| 59 | \( 1 + 2.58T + 59T^{2} \) |
| 61 | \( 1 - 3.71T + 61T^{2} \) |
| 67 | \( 1 + 8.86T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 9.44T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 - 1.75T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
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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
−7.41071343695879083958880753538, −6.72424136306248245161010785238, −6.11729602317828646721622645859, −5.56965376723155833151075930337, −4.67584007102615375864792178436, −3.87119923335918470657531533710, −3.57078884783953226666904571460, −2.27550500066033274202694183598, −1.37524699536699559647608680968, 0,
1.37524699536699559647608680968, 2.27550500066033274202694183598, 3.57078884783953226666904571460, 3.87119923335918470657531533710, 4.67584007102615375864792178436, 5.56965376723155833151075930337, 6.11729602317828646721622645859, 6.72424136306248245161010785238, 7.41071343695879083958880753538