| L(s) = 1 | + 21·5-s − 8·7-s − 36·11-s − 49·13-s + 21·17-s + 112·19-s − 180·23-s + 316·25-s − 135·29-s − 308·31-s − 168·35-s − 37-s − 42·41-s − 20·43-s − 84·47-s − 279·49-s − 174·53-s − 756·55-s − 504·59-s − 385·61-s − 1.02e3·65-s − 272·67-s + 888·71-s + 371·73-s + 288·77-s + 652·79-s − 84·83-s + ⋯ |
| L(s) = 1 | + 1.87·5-s − 0.431·7-s − 0.986·11-s − 1.04·13-s + 0.299·17-s + 1.35·19-s − 1.63·23-s + 2.52·25-s − 0.864·29-s − 1.78·31-s − 0.811·35-s − 0.00444·37-s − 0.159·41-s − 0.0709·43-s − 0.260·47-s − 0.813·49-s − 0.450·53-s − 1.85·55-s − 1.11·59-s − 0.808·61-s − 1.96·65-s − 0.495·67-s + 1.48·71-s + 0.594·73-s + 0.426·77-s + 0.928·79-s − 0.111·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 21 T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 49 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 112 T + p^{3} T^{2} \) |
| 23 | \( 1 + 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 135 T + p^{3} T^{2} \) |
| 31 | \( 1 + 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 + 20 T + p^{3} T^{2} \) |
| 47 | \( 1 + 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 504 T + p^{3} T^{2} \) |
| 61 | \( 1 + 385 T + p^{3} T^{2} \) |
| 67 | \( 1 + 272 T + p^{3} T^{2} \) |
| 71 | \( 1 - 888 T + p^{3} T^{2} \) |
| 73 | \( 1 - 371 T + p^{3} T^{2} \) |
| 79 | \( 1 - 652 T + p^{3} T^{2} \) |
| 83 | \( 1 + 84 T + p^{3} T^{2} \) |
| 89 | \( 1 - 21 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1246 T + p^{3} T^{2} \) |
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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
−9.366536543368594832795556620373, −7.970967483417211027528953341778, −7.22857585039513968330156268356, −6.19683100561735671975964392744, −5.51390720421734857603560079577, −4.98625952190502882821324636342, −3.37805623070009378767628409863, −2.40715897410830656489103212031, −1.64058532493892929907620479227, 0,
1.64058532493892929907620479227, 2.40715897410830656489103212031, 3.37805623070009378767628409863, 4.98625952190502882821324636342, 5.51390720421734857603560079577, 6.19683100561735671975964392744, 7.22857585039513968330156268356, 7.970967483417211027528953341778, 9.366536543368594832795556620373
