L(s) = 1 | − 5-s − 2·11-s − 13-s − 2·17-s + 6·19-s + 4·23-s + 25-s + 8·29-s − 8·31-s + 2·37-s + 6·41-s − 4·43-s + 4·47-s − 6·53-s + 2·55-s + 12·61-s + 65-s − 6·67-s − 12·71-s + 6·73-s + 6·83-s + 2·85-s + 6·89-s − 6·95-s − 14·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.603·11-s − 0.277·13-s − 0.485·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s + 1.48·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.583·47-s − 0.824·53-s + 0.269·55-s + 1.53·61-s + 0.124·65-s − 0.733·67-s − 1.42·71-s + 0.702·73-s + 0.658·83-s + 0.216·85-s + 0.635·89-s − 0.615·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.11527748138325, −12.72014415815013, −12.25193613156827, −11.69847311751142, −11.39066656425211, −10.83960867695952, −10.43026942988554, −9.955825723164384, −9.275510549222184, −9.090493549530351, −8.420206900894132, −7.895997875390625, −7.541977491035282, −6.997960371515373, −6.658208817148745, −5.853654525844027, −5.460218378682238, −4.847128563712251, −4.560765947744404, −3.782826492700428, −3.281429656532714, −2.745862614465912, −2.252873066456787, −1.358128546563839, −0.7912433043592371, 0,
0.7912433043592371, 1.358128546563839, 2.252873066456787, 2.745862614465912, 3.281429656532714, 3.782826492700428, 4.560765947744404, 4.847128563712251, 5.460218378682238, 5.853654525844027, 6.658208817148745, 6.997960371515373, 7.541977491035282, 7.895997875390625, 8.420206900894132, 9.090493549530351, 9.275510549222184, 9.955825723164384, 10.43026942988554, 10.83960867695952, 11.39066656425211, 11.69847311751142, 12.25193613156827, 12.72014415815013, 13.11527748138325