L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 3·13-s + 6·17-s − 3·19-s − 21-s + 5·23-s − 27-s − 7·29-s + 3·31-s + 33-s − 4·37-s − 3·39-s − 8·41-s + 13·43-s + 49-s − 6·51-s + 12·53-s + 3·57-s + 8·59-s − 10·61-s + 63-s + 12·67-s − 5·69-s − 5·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 1.45·17-s − 0.688·19-s − 0.218·21-s + 1.04·23-s − 0.192·27-s − 1.29·29-s + 0.538·31-s + 0.174·33-s − 0.657·37-s − 0.480·39-s − 1.24·41-s + 1.98·43-s + 1/7·49-s − 0.840·51-s + 1.64·53-s + 0.397·57-s + 1.04·59-s − 1.28·61-s + 0.125·63-s + 1.46·67-s − 0.601·69-s − 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.93568331974353, −13.61110691334828, −13.09897919487133, −12.43477247857437, −12.26259793393521, −11.55359247018623, −11.09684985862575, −10.68408895265378, −10.29591319220165, −9.636763852439332, −9.136296019039590, −8.513872464180710, −8.079387121702900, −7.497210954418457, −6.968894208534039, −6.474917875934511, −5.677387940960073, −5.465908939493901, −4.960302812236064, −4.016742064695732, −3.829175582867647, −2.958632938753029, −2.312413289206777, −1.403071782354198, −1.003111811896128, 0,
1.003111811896128, 1.403071782354198, 2.312413289206777, 2.958632938753029, 3.829175582867647, 4.016742064695732, 4.960302812236064, 5.465908939493901, 5.677387940960073, 6.474917875934511, 6.968894208534039, 7.497210954418457, 8.079387121702900, 8.513872464180710, 9.136296019039590, 9.636763852439332, 10.29591319220165, 10.68408895265378, 11.09684985862575, 11.55359247018623, 12.26259793393521, 12.43477247857437, 13.09897919487133, 13.61110691334828, 13.93568331974353