L(s) = 1 | − 5-s − 7-s − 2·11-s − 13-s + 2·17-s + 2·19-s + 23-s + 25-s − 5·29-s − 5·31-s + 35-s + 8·37-s − 5·41-s + 10·43-s − 47-s + 49-s + 2·55-s + 4·59-s − 8·61-s + 65-s − 8·67-s − 71-s − 7·73-s + 2·77-s + 14·79-s − 6·83-s − 2·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.603·11-s − 0.277·13-s + 0.485·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 0.928·29-s − 0.898·31-s + 0.169·35-s + 1.31·37-s − 0.780·41-s + 1.52·43-s − 0.145·47-s + 1/7·49-s + 0.269·55-s + 0.520·59-s − 1.02·61-s + 0.124·65-s − 0.977·67-s − 0.118·71-s − 0.819·73-s + 0.227·77-s + 1.57·79-s − 0.658·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.77925386413898, −13.22299112435051, −12.98150577834470, −12.32401454205539, −12.00355642821682, −11.43907057973315, −10.80350307938842, −10.63360531334998, −9.850603945186615, −9.431200864259832, −9.069170546711621, −8.329557166369106, −7.847377268090634, −7.341855902672303, −7.118020449040969, −6.191896805694843, −5.830070218223264, −5.234209318763313, −4.701802050257762, −4.043261304405336, −3.501270344443919, −2.937325404599521, −2.375114399486425, −1.573608599102746, −0.7537584918059707, 0,
0.7537584918059707, 1.573608599102746, 2.375114399486425, 2.937325404599521, 3.501270344443919, 4.043261304405336, 4.701802050257762, 5.234209318763313, 5.830070218223264, 6.191896805694843, 7.118020449040969, 7.341855902672303, 7.847377268090634, 8.329557166369106, 9.069170546711621, 9.431200864259832, 9.850603945186615, 10.63360531334998, 10.80350307938842, 11.43907057973315, 12.00355642821682, 12.32401454205539, 12.98150577834470, 13.22299112435051, 13.77925386413898