| L(s) = 1 | + 5-s − 2·7-s − 2·11-s + 2·17-s + 4·19-s + 25-s + 2·29-s − 8·31-s − 2·35-s + 4·37-s + 8·41-s + 8·43-s − 8·47-s − 3·49-s + 10·53-s − 2·55-s + 6·59-s − 2·61-s + 12·67-s + 12·71-s − 2·73-s + 4·77-s − 8·79-s + 4·83-s + 2·85-s + 12·89-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.603·11-s + 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.338·35-s + 0.657·37-s + 1.24·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 1.37·53-s − 0.269·55-s + 0.781·59-s − 0.256·61-s + 1.46·67-s + 1.42·71-s − 0.234·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s + 0.216·85-s + 1.27·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.728393883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728393883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.918477822601378665400349692738, −7.909872736408381331262556551925, −7.31613585971718924467229604873, −6.44264161666913481060497810496, −5.68299655011922972484706633320, −5.07904235011738772379730992224, −3.90516997133058074350955741790, −3.07395237449891549483690781139, −2.19505322059612106828769223593, −0.805470072675148432900611802709,
0.805470072675148432900611802709, 2.19505322059612106828769223593, 3.07395237449891549483690781139, 3.90516997133058074350955741790, 5.07904235011738772379730992224, 5.68299655011922972484706633320, 6.44264161666913481060497810496, 7.31613585971718924467229604873, 7.909872736408381331262556551925, 8.918477822601378665400349692738
