| L(s) = 1 | − 3-s + 3·5-s + 2·7-s − 2·9-s + 11-s + 4·13-s − 3·15-s + 6·17-s − 8·19-s − 2·21-s − 3·23-s + 4·25-s + 5·27-s + 5·31-s − 33-s + 6·35-s + 37-s − 4·39-s + 10·43-s − 6·45-s − 3·49-s − 6·51-s + 6·53-s + 3·55-s + 8·57-s − 3·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s + 1.10·13-s − 0.774·15-s + 1.45·17-s − 1.83·19-s − 0.436·21-s − 0.625·23-s + 4/5·25-s + 0.962·27-s + 0.898·31-s − 0.174·33-s + 1.01·35-s + 0.164·37-s − 0.640·39-s + 1.52·43-s − 0.894·45-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.404·55-s + 1.05·57-s − 0.390·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706912539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706912539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−10.54980513640919233558709119533, −9.700092895780422192241801282926, −8.674866558231347995249063814171, −8.050490063857036795242146503456, −6.51356460177744683595774086764, −5.96029809065759357817228429363, −5.29396314589658220886171477329, −4.04654564288201609443200871032, −2.49141172081529131249029798363, −1.28296422834420783986833535835,
1.28296422834420783986833535835, 2.49141172081529131249029798363, 4.04654564288201609443200871032, 5.29396314589658220886171477329, 5.96029809065759357817228429363, 6.51356460177744683595774086764, 8.050490063857036795242146503456, 8.674866558231347995249063814171, 9.700092895780422192241801282926, 10.54980513640919233558709119533
