L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s + 4·11-s − 13-s + 2·15-s + 2·17-s − 4·19-s − 21-s + 4·23-s − 25-s + 27-s + 10·29-s − 8·31-s + 4·33-s − 2·35-s + 2·37-s − 39-s + 10·41-s − 4·43-s + 2·45-s + 8·47-s + 49-s + 2·51-s − 6·53-s + 8·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.696·33-s − 0.338·35-s + 0.328·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.080426503\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.080426503\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.591909054463627013254143334549, −7.60737624627539231014495398201, −6.83423409318481332335160340857, −6.27123411237532440751009381950, −5.52313078018151925153558509277, −4.51035263133546415298573729927, −3.76910138393662928748214031195, −2.82842048602009961166929070702, −2.02297835901821089070845609831, −1.01235796598496708925518877535,
1.01235796598496708925518877535, 2.02297835901821089070845609831, 2.82842048602009961166929070702, 3.76910138393662928748214031195, 4.51035263133546415298573729927, 5.52313078018151925153558509277, 6.27123411237532440751009381950, 6.83423409318481332335160340857, 7.60737624627539231014495398201, 8.591909054463627013254143334549