| L(s) = 1 | + 2·5-s − 4·11-s + 6·13-s + 6·17-s − 25-s − 6·29-s + 4·31-s + 37-s + 2·41-s + 4·43-s + 8·47-s + 10·53-s − 8·55-s + 8·59-s − 2·61-s + 12·65-s − 4·67-s − 8·71-s + 6·73-s − 8·79-s + 12·83-s + 12·85-s + 14·89-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s + 1.66·13-s + 1.45·17-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.164·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1.37·53-s − 1.07·55-s + 1.04·59-s − 0.256·61-s + 1.48·65-s − 0.488·67-s − 0.949·71-s + 0.702·73-s − 0.900·79-s + 1.31·83-s + 1.30·85-s + 1.48·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.862098965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.862098965\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.40950579772861, −13.14420738951023, −12.70998058000314, −11.92847506040276, −11.67791972640006, −10.85588675659616, −10.58800601091639, −10.15225575248191, −9.689860581074882, −9.039051951947750, −8.716874163188778, −7.965626860078529, −7.702495486116348, −7.129040652567924, −6.286728460769207, −5.942772940976576, −5.523072266352737, −5.171645649363368, −4.255699607794918, −3.732802172816909, −3.168992041823315, −2.495659778574527, −1.958145704817107, −1.188640092653400, −0.6423592793139160,
0.6423592793139160, 1.188640092653400, 1.958145704817107, 2.495659778574527, 3.168992041823315, 3.732802172816909, 4.255699607794918, 5.171645649363368, 5.523072266352737, 5.942772940976576, 6.286728460769207, 7.129040652567924, 7.702495486116348, 7.965626860078529, 8.716874163188778, 9.039051951947750, 9.689860581074882, 10.15225575248191, 10.58800601091639, 10.85588675659616, 11.67791972640006, 11.92847506040276, 12.70998058000314, 13.14420738951023, 13.40950579772861
