L(s) = 1 | − 5-s + 11-s − 2·13-s − 7·17-s − 3·19-s − 3·23-s + 25-s + 5·29-s + 2·37-s + 43-s + 8·47-s + 9·53-s − 55-s − 9·59-s − 5·61-s + 2·65-s − 2·67-s + 12·71-s − 14·79-s + 9·83-s + 7·85-s + 5·89-s + 3·95-s − 7·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.301·11-s − 0.554·13-s − 1.69·17-s − 0.688·19-s − 0.625·23-s + 1/5·25-s + 0.928·29-s + 0.328·37-s + 0.152·43-s + 1.16·47-s + 1.23·53-s − 0.134·55-s − 1.17·59-s − 0.640·61-s + 0.248·65-s − 0.244·67-s + 1.42·71-s − 1.57·79-s + 0.987·83-s + 0.759·85-s + 0.529·89-s + 0.307·95-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.92209181688953, −13.63756382656552, −13.03044951056217, −12.46538042041295, −12.14187041615984, −11.59112170058487, −11.08683967984753, −10.62703354668001, −10.17482353698310, −9.530409203582620, −8.934683335913607, −8.657938769480787, −8.050988131952017, −7.449956623297515, −7.007372477721609, −6.358865898458310, −6.095588335040474, −5.196224317636048, −4.642732863900845, −4.191133094811268, −3.757531141446516, −2.801238236311056, −2.384865174160570, −1.716552816788143, −0.7402739988655993, 0,
0.7402739988655993, 1.716552816788143, 2.384865174160570, 2.801238236311056, 3.757531141446516, 4.191133094811268, 4.642732863900845, 5.196224317636048, 6.095588335040474, 6.358865898458310, 7.007372477721609, 7.449956623297515, 8.050988131952017, 8.657938769480787, 8.934683335913607, 9.530409203582620, 10.17482353698310, 10.62703354668001, 11.08683967984753, 11.59112170058487, 12.14187041615984, 12.46538042041295, 13.03044951056217, 13.63756382656552, 13.92209181688953