L(s) = 1 | + 4·5-s − 2·7-s + 4·11-s + 2·13-s + 2·17-s − 8·19-s + 4·23-s + 11·25-s + 6·31-s − 8·35-s − 2·37-s − 6·41-s + 4·47-s − 3·49-s + 16·55-s − 4·59-s + 14·61-s + 8·65-s − 4·67-s + 12·71-s − 10·73-s − 8·77-s − 10·79-s − 12·83-s + 8·85-s + 14·89-s − 4·91-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 0.755·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 1.83·19-s + 0.834·23-s + 11/5·25-s + 1.07·31-s − 1.35·35-s − 0.328·37-s − 0.937·41-s + 0.583·47-s − 3/7·49-s + 2.15·55-s − 0.520·59-s + 1.79·61-s + 0.992·65-s − 0.488·67-s + 1.42·71-s − 1.17·73-s − 0.911·77-s − 1.12·79-s − 1.31·83-s + 0.867·85-s + 1.48·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.261308022\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.261308022\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.872147170802695777042819267450, −9.003322740980815328578766143835, −8.551716675388505933005243425858, −6.89547356553480728554873620104, −6.39174188210860038385829426400, −5.83450410447989994046324048602, −4.69490327290429029526959386590, −3.49343359434470738584054327342, −2.35029669656051858459889649700, −1.28088417651653847704023663777,
1.28088417651653847704023663777, 2.35029669656051858459889649700, 3.49343359434470738584054327342, 4.69490327290429029526959386590, 5.83450410447989994046324048602, 6.39174188210860038385829426400, 6.89547356553480728554873620104, 8.551716675388505933005243425858, 9.003322740980815328578766143835, 9.872147170802695777042819267450