L(s) = 1 | + 5-s − 3·9-s + 11-s + 6·13-s + 17-s − 2·19-s − 2·23-s + 25-s − 2·29-s + 4·31-s + 2·37-s + 12·41-s − 8·43-s − 3·45-s + 8·47-s − 7·49-s − 4·53-s + 55-s − 12·59-s + 10·61-s + 6·65-s − 2·67-s − 8·71-s + 2·73-s + 10·79-s + 9·81-s + 16·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 9-s + 0.301·11-s + 1.66·13-s + 0.242·17-s − 0.458·19-s − 0.417·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.328·37-s + 1.87·41-s − 1.21·43-s − 0.447·45-s + 1.16·47-s − 49-s − 0.549·53-s + 0.134·55-s − 1.56·59-s + 1.28·61-s + 0.744·65-s − 0.244·67-s − 0.949·71-s + 0.234·73-s + 1.12·79-s + 81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.648830597\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.648830597\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.11139044325589, −13.92765338948451, −13.36601889429006, −12.89649023460876, −12.27173697028259, −11.73379302254203, −11.19117750211260, −10.84094609159292, −10.33556819770385, −9.602527442726900, −9.078557732938159, −8.761493848085341, −7.987063014983376, −7.847308879067652, −6.764643990921798, −6.256482594734576, −6.002003765437194, −5.405639516215179, −4.674044257404618, −3.977576091681569, −3.433017710264939, −2.796773250608127, −2.076277021878212, −1.335153404835019, −0.5800494440579943,
0.5800494440579943, 1.335153404835019, 2.076277021878212, 2.796773250608127, 3.433017710264939, 3.977576091681569, 4.674044257404618, 5.405639516215179, 6.002003765437194, 6.256482594734576, 6.764643990921798, 7.847308879067652, 7.987063014983376, 8.761493848085341, 9.078557732938159, 9.602527442726900, 10.33556819770385, 10.84094609159292, 11.19117750211260, 11.73379302254203, 12.27173697028259, 12.89649023460876, 13.36601889429006, 13.92765338948451, 14.11139044325589