L(s) = 1 | − 5-s + 11-s − 2·13-s − 3·17-s − 7·19-s − 3·23-s + 25-s + 3·29-s − 4·31-s − 10·37-s + 7·43-s − 9·53-s − 55-s − 3·59-s + 61-s + 2·65-s − 2·67-s − 12·71-s + 4·73-s − 2·79-s + 3·83-s + 3·85-s + 9·89-s + 7·95-s + 19·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.301·11-s − 0.554·13-s − 0.727·17-s − 1.60·19-s − 0.625·23-s + 1/5·25-s + 0.557·29-s − 0.718·31-s − 1.64·37-s + 1.06·43-s − 1.23·53-s − 0.134·55-s − 0.390·59-s + 0.128·61-s + 0.248·65-s − 0.244·67-s − 1.42·71-s + 0.468·73-s − 0.225·79-s + 0.329·83-s + 0.325·85-s + 0.953·89-s + 0.718·95-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−12.61095207173790, −12.12668078717587, −11.98339446962800, −11.26041631948307, −10.88973899257152, −10.43904275197832, −10.16328292916062, −9.352439997938007, −9.094512944934460, −8.555658848922892, −8.252735210652422, −7.568301223931862, −7.254653335940992, −6.694199236209839, −6.229739151722453, −5.884506745875283, −5.087887677143731, −4.624906197356781, −4.317845389017622, −3.669609952466283, −3.268834076448109, −2.470496898927349, −2.046577715832921, −1.532222353973838, −0.5505035573704109, 0,
0.5505035573704109, 1.532222353973838, 2.046577715832921, 2.470496898927349, 3.268834076448109, 3.669609952466283, 4.317845389017622, 4.624906197356781, 5.087887677143731, 5.884506745875283, 6.229739151722453, 6.694199236209839, 7.254653335940992, 7.568301223931862, 8.252735210652422, 8.555658848922892, 9.094512944934460, 9.352439997938007, 10.16328292916062, 10.43904275197832, 10.88973899257152, 11.26041631948307, 11.98339446962800, 12.12668078717587, 12.61095207173790