L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 4·11-s − 5·13-s − 15-s + 2·17-s + 6·19-s − 2·21-s + 6·23-s + 25-s − 27-s − 3·29-s − 6·31-s + 4·33-s + 2·35-s + 5·39-s − 11·43-s + 45-s − 9·47-s − 3·49-s − 2·51-s + 3·53-s − 4·55-s − 6·57-s − 59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.38·13-s − 0.258·15-s + 0.485·17-s + 1.37·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 1.07·31-s + 0.696·33-s + 0.338·35-s + 0.800·39-s − 1.67·43-s + 0.149·45-s − 1.31·47-s − 3/7·49-s − 0.280·51-s + 0.412·53-s − 0.539·55-s − 0.794·57-s − 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−12.81947293891176, −12.37054225062112, −11.90215206952151, −11.33561949083903, −11.19006960661643, −10.50877305046953, −10.15380631673767, −9.597952329306940, −9.472190785782595, −8.653917514862811, −8.165147308980301, −7.604399844398440, −7.334243672453355, −6.934289422706529, −6.242072112075516, −5.567014931402343, −5.224323506776360, −4.930957421173382, −4.684260613058712, −3.567851389837212, −3.236434373748331, −2.567375057949857, −1.974359775475281, −1.473649886633268, −0.7156839371096786, 0,
0.7156839371096786, 1.473649886633268, 1.974359775475281, 2.567375057949857, 3.236434373748331, 3.567851389837212, 4.684260613058712, 4.930957421173382, 5.224323506776360, 5.567014931402343, 6.242072112075516, 6.934289422706529, 7.334243672453355, 7.604399844398440, 8.165147308980301, 8.653917514862811, 9.472190785782595, 9.597952329306940, 10.15380631673767, 10.50877305046953, 11.19006960661643, 11.33561949083903, 11.90215206952151, 12.37054225062112, 12.81947293891176