| L(s) = 1 | − 3-s − 2·7-s + 9-s − 6·11-s + 13-s + 17-s − 2·19-s + 2·21-s + 6·23-s − 27-s − 9·29-s + 9·31-s + 6·33-s + 12·37-s − 39-s + 6·41-s − 6·43-s + 12·47-s − 3·49-s − 51-s + 3·53-s + 2·57-s − 7·59-s − 6·61-s − 2·63-s + 13·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 0.242·17-s − 0.458·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s − 1.67·29-s + 1.61·31-s + 1.04·33-s + 1.97·37-s − 0.160·39-s + 0.937·41-s − 0.914·43-s + 1.75·47-s − 3/7·49-s − 0.140·51-s + 0.412·53-s + 0.264·57-s − 0.911·59-s − 0.768·61-s − 0.251·63-s + 1.58·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−13.13576771630789, −12.63543649938039, −12.22055423575017, −11.51863285682634, −11.20154444370061, −10.61327149353812, −10.44497448131332, −9.847176089271771, −9.381717424245257, −8.971564492843226, −8.268115757757950, −7.757673432196366, −7.504162687028288, −6.855285717558146, −6.350218615186113, −5.797211131292414, −5.565563673065388, −4.884243884664979, −4.445229498585154, −3.883924396463720, −3.066746061940584, −2.768460313723376, −2.230227494222567, −1.298557861280963, −0.6483863282717300, 0,
0.6483863282717300, 1.298557861280963, 2.230227494222567, 2.768460313723376, 3.066746061940584, 3.883924396463720, 4.445229498585154, 4.884243884664979, 5.565563673065388, 5.797211131292414, 6.350218615186113, 6.855285717558146, 7.504162687028288, 7.757673432196366, 8.268115757757950, 8.971564492843226, 9.381717424245257, 9.847176089271771, 10.44497448131332, 10.61327149353812, 11.20154444370061, 11.51863285682634, 12.22055423575017, 12.63543649938039, 13.13576771630789
