| 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
−8.943789459903209121383830188153, −7.940723245377987317077690617612, −7.17485784174274370950965843414, −6.81991635983329139306063307616, −5.30325988333506715807666894305, −4.47697502960724786319707485114, −3.60126964613226170557300052100, −2.90204035794661186908030892407, −1.07567772053420290956943714256, 0,
1.07567772053420290956943714256, 2.90204035794661186908030892407, 3.60126964613226170557300052100, 4.47697502960724786319707485114, 5.30325988333506715807666894305, 6.81991635983329139306063307616, 7.17485784174274370950965843414, 7.940723245377987317077690617612, 8.943789459903209121383830188153
