L(s) = 1 | − 5-s − 7-s + 4·11-s − 2·13-s + 6·17-s − 4·19-s + 8·23-s + 25-s + 2·29-s + 35-s − 2·37-s − 10·41-s − 4·43-s + 49-s − 14·53-s − 4·55-s + 12·59-s − 2·61-s + 2·65-s + 4·67-s + 2·73-s − 4·77-s + 8·79-s − 4·83-s − 6·85-s + 6·89-s + 2·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 0.169·35-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 1/7·49-s − 1.92·53-s − 0.539·55-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.234·73-s − 0.455·77-s + 0.900·79-s − 0.439·83-s − 0.650·85-s + 0.635·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.764360525\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.764360525\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.341504705813896398806804082481, −7.40416170019680941269110344661, −6.83103018167217491535952196628, −6.23140136123531996220057035320, −5.20958257504061441606615609435, −4.58418850262270517583381292844, −3.55775255722757430610867518948, −3.12411412832318289993654168292, −1.81064313124971820952332684962, −0.74216667907756560942015911650,
0.74216667907756560942015911650, 1.81064313124971820952332684962, 3.12411412832318289993654168292, 3.55775255722757430610867518948, 4.58418850262270517583381292844, 5.20958257504061441606615609435, 6.23140136123531996220057035320, 6.83103018167217491535952196628, 7.40416170019680941269110344661, 8.341504705813896398806804082481