| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 9·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 18·10-s − 18·11-s − 12·12-s + 13·13-s + 14·14-s + 27·15-s + 16·16-s + 60·17-s + 18·18-s − 43·19-s − 36·20-s − 21·21-s − 36·22-s + 9·23-s − 24·24-s − 44·25-s + 26·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.804·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.569·10-s − 0.493·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.464·15-s + 1/4·16-s + 0.856·17-s + 0.235·18-s − 0.519·19-s − 0.402·20-s − 0.218·21-s − 0.348·22-s + 0.0815·23-s − 0.204·24-s − 0.351·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 11 | \( 1 + 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 60 T + p^{3} T^{2} \) |
| 19 | \( 1 + 43 T + p^{3} T^{2} \) |
| 23 | \( 1 - 9 T + p^{3} T^{2} \) |
| 29 | \( 1 + 249 T + p^{3} T^{2} \) |
| 31 | \( 1 + 79 T + p^{3} T^{2} \) |
| 37 | \( 1 + 412 T + p^{3} T^{2} \) |
| 41 | \( 1 - 222 T + p^{3} T^{2} \) |
| 43 | \( 1 + 295 T + p^{3} T^{2} \) |
| 47 | \( 1 - 411 T + p^{3} T^{2} \) |
| 53 | \( 1 + 237 T + p^{3} T^{2} \) |
| 59 | \( 1 + 384 T + p^{3} T^{2} \) |
| 61 | \( 1 + 466 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1042 T + p^{3} T^{2} \) |
| 71 | \( 1 + 288 T + p^{3} T^{2} \) |
| 73 | \( 1 + 691 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1001 T + p^{3} T^{2} \) |
| 83 | \( 1 - 39 T + p^{3} T^{2} \) |
| 89 | \( 1 + 339 T + p^{3} T^{2} \) |
| 97 | \( 1 - 713 T + p^{3} T^{2} \) |
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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
−10.36440185803305574355754140721, −9.023061635789379752255093794511, −7.82545494426059167413160200851, −7.29291056210398122820866510793, −6.03049398110559321558802236987, −5.25434251724776577676065199842, −4.23850565445440865924778209217, −3.32320653048887885062754673879, −1.70353823275497890748267666361, 0,
1.70353823275497890748267666361, 3.32320653048887885062754673879, 4.23850565445440865924778209217, 5.25434251724776577676065199842, 6.03049398110559321558802236987, 7.29291056210398122820866510793, 7.82545494426059167413160200851, 9.023061635789379752255093794511, 10.36440185803305574355754140721
