| L(s) = 1 | + 2-s − 4-s − 2·5-s + 7-s − 3·8-s − 2·10-s + 2·13-s + 14-s − 16-s − 2·17-s + 2·20-s + 8·23-s − 25-s + 2·26-s − 28-s − 29-s + 8·31-s + 5·32-s − 2·34-s − 2·35-s − 10·37-s + 6·40-s + 6·41-s − 4·43-s + 8·46-s − 8·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.06·8-s − 0.632·10-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.447·20-s + 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.185·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s − 0.338·35-s − 1.64·37-s + 0.948·40-s + 0.937·41-s − 0.609·43-s + 1.17·46-s − 1.16·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.118320892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118320892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.95941411223670, −12.55007602091958, −12.08486633580825, −11.68465659633247, −11.10673112010813, −10.92898816343916, −10.22785129340286, −9.546958253952118, −9.221763848691819, −8.643624465645973, −8.175750575576071, −7.979678151961419, −7.135578790687587, −6.707428815425471, −6.258425282364558, −5.524393769778360, −5.135656439445232, −4.557094455703096, −4.278967435153747, −3.681827327355622, −3.054326744103598, −2.849294776378736, −1.737056827696230, −1.153929802537112, −0.2834423582652990,
0.2834423582652990, 1.153929802537112, 1.737056827696230, 2.849294776378736, 3.054326744103598, 3.681827327355622, 4.278967435153747, 4.557094455703096, 5.135656439445232, 5.524393769778360, 6.258425282364558, 6.707428815425471, 7.135578790687587, 7.979678151961419, 8.175750575576071, 8.643624465645973, 9.221763848691819, 9.546958253952118, 10.22785129340286, 10.92898816343916, 11.10673112010813, 11.68465659633247, 12.08486633580825, 12.55007602091958, 12.95941411223670
