L(s) = 1 | + 2·5-s + 7-s − 11-s − 6·13-s − 5·17-s + 7·19-s − 4·23-s − 25-s − 4·29-s + 6·31-s + 2·35-s + 2·37-s + 3·41-s + 43-s + 49-s + 12·53-s − 2·55-s + 7·59-s − 12·61-s − 12·65-s − 13·67-s + 8·71-s + 73-s − 77-s + 6·79-s − 16·83-s − 10·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.301·11-s − 1.66·13-s − 1.21·17-s + 1.60·19-s − 0.834·23-s − 1/5·25-s − 0.742·29-s + 1.07·31-s + 0.338·35-s + 0.328·37-s + 0.468·41-s + 0.152·43-s + 1/7·49-s + 1.64·53-s − 0.269·55-s + 0.911·59-s − 1.53·61-s − 1.48·65-s − 1.58·67-s + 0.949·71-s + 0.117·73-s − 0.113·77-s + 0.675·79-s − 1.75·83-s − 1.08·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.42730868418561709330643460317, −6.76786157608618491132092170330, −5.87917024304656084099884427091, −5.36806250547312900643600739722, −4.70686874201751041174943906105, −3.97333715195737154416319433517, −2.69117835651894179668423576360, −2.35201703266852461460726578003, −1.35975274929503914135303042484, 0,
1.35975274929503914135303042484, 2.35201703266852461460726578003, 2.69117835651894179668423576360, 3.97333715195737154416319433517, 4.70686874201751041174943906105, 5.36806250547312900643600739722, 5.87917024304656084099884427091, 6.76786157608618491132092170330, 7.42730868418561709330643460317