L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s − 6·13-s − 16-s − 6·17-s − 19-s − 2·20-s + 4·23-s − 25-s + 6·26-s + 2·29-s − 5·32-s + 6·34-s + 10·37-s + 38-s + 6·40-s + 2·41-s + 4·43-s − 4·46-s − 12·47-s − 7·49-s + 50-s + 6·52-s − 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s − 1.66·13-s − 1/4·16-s − 1.45·17-s − 0.229·19-s − 0.447·20-s + 0.834·23-s − 1/5·25-s + 1.17·26-s + 0.371·29-s − 0.883·32-s + 1.02·34-s + 1.64·37-s + 0.162·38-s + 0.948·40-s + 0.312·41-s + 0.609·43-s − 0.589·46-s − 1.75·47-s − 49-s + 0.141·50-s + 0.832·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.19520842735944, −13.15398431533369, −12.81825201162161, −12.03587374319331, −11.43135787046076, −11.08611698218034, −10.38562460289057, −10.08409850955177, −9.580996955906052, −9.223995549031246, −8.953084353322882, −8.113899694766386, −7.892414407646687, −7.214895077160906, −6.721171656293649, −6.264644628129849, −5.570695235022525, −4.938797960085733, −4.654246763426405, −4.196645547129962, −3.285682419150437, −2.533290252334768, −2.160303524507141, −1.523561212386859, −0.6806416220616605, 0,
0.6806416220616605, 1.523561212386859, 2.160303524507141, 2.533290252334768, 3.285682419150437, 4.196645547129962, 4.654246763426405, 4.938797960085733, 5.570695235022525, 6.264644628129849, 6.721171656293649, 7.214895077160906, 7.892414407646687, 8.113899694766386, 8.953084353322882, 9.223995549031246, 9.580996955906052, 10.08409850955177, 10.38562460289057, 11.08611698218034, 11.43135787046076, 12.03587374319331, 12.81825201162161, 13.15398431533369, 13.19520842735944