| L(s) = 1 | + 3-s − 4·7-s + 9-s + 6·17-s + 19-s − 4·21-s + 8·23-s + 27-s + 2·29-s − 8·37-s + 6·43-s + 9·49-s + 6·51-s − 6·53-s + 57-s + 10·59-s − 14·61-s − 4·63-s + 12·67-s + 8·69-s + 4·71-s + 81-s − 4·83-s + 2·87-s + 4·89-s − 14·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.45·17-s + 0.229·19-s − 0.872·21-s + 1.66·23-s + 0.192·27-s + 0.371·29-s − 1.31·37-s + 0.914·43-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 0.132·57-s + 1.30·59-s − 1.79·61-s − 0.503·63-s + 1.46·67-s + 0.963·69-s + 0.474·71-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + 0.423·89-s − 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.394090897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394090897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.62434921592565, −14.92240621082143, −14.39471578096643, −13.86595045157813, −13.31545222468949, −12.79527031900074, −12.35806550649296, −11.91006685536632, −10.91657825902316, −10.52861378739684, −9.772527880068358, −9.489311416361855, −8.964554449401856, −8.237470449461273, −7.653049665003957, −6.841946623718194, −6.727767560924861, −5.693275231714577, −5.293007422969804, −4.349177672692340, −3.526343114374230, −3.150182601573691, −2.615136004466298, −1.466941475253129, −0.6276098998082903,
0.6276098998082903, 1.466941475253129, 2.615136004466298, 3.150182601573691, 3.526343114374230, 4.349177672692340, 5.293007422969804, 5.693275231714577, 6.727767560924861, 6.841946623718194, 7.653049665003957, 8.237470449461273, 8.964554449401856, 9.489311416361855, 9.772527880068358, 10.52861378739684, 10.91657825902316, 11.91006685536632, 12.35806550649296, 12.79527031900074, 13.31545222468949, 13.86595045157813, 14.39471578096643, 14.92240621082143, 15.62434921592565
