| L(s) = 1 | + 3·3-s + 2·5-s − 7-s + 6·9-s − 11-s − 4·13-s + 6·15-s − 2·17-s − 7·19-s − 3·21-s − 9·23-s − 25-s + 9·27-s − 29-s − 3·33-s − 2·35-s + 8·37-s − 12·39-s + 41-s + 2·43-s + 12·45-s + 3·47-s + 49-s − 6·51-s − 9·53-s − 2·55-s − 21·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 0.377·7-s + 2·9-s − 0.301·11-s − 1.10·13-s + 1.54·15-s − 0.485·17-s − 1.60·19-s − 0.654·21-s − 1.87·23-s − 1/5·25-s + 1.73·27-s − 0.185·29-s − 0.522·33-s − 0.338·35-s + 1.31·37-s − 1.92·39-s + 0.156·41-s + 0.304·43-s + 1.78·45-s + 0.437·47-s + 1/7·49-s − 0.840·51-s − 1.23·53-s − 0.269·55-s − 2.78·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.590418569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.590418569\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.43353773926105, −13.03608772973673, −12.64638130283728, −12.24421142845438, −11.48432103998255, −10.77962201040569, −10.22582854171676, −9.920530350465249, −9.433806746869016, −9.216843991789513, −8.428306875056478, −8.171668288001766, −7.641496247505865, −7.153220100026760, −6.457177404127087, −6.096185173353889, −5.467520712592219, −4.573971883562294, −4.243240125933441, −3.730998360174842, −2.919066314529307, −2.444502508121245, −2.081147778595812, −1.703858422241835, −0.4424824502252229,
0.4424824502252229, 1.703858422241835, 2.081147778595812, 2.444502508121245, 2.919066314529307, 3.730998360174842, 4.243240125933441, 4.573971883562294, 5.467520712592219, 6.096185173353889, 6.457177404127087, 7.153220100026760, 7.641496247505865, 8.171668288001766, 8.428306875056478, 9.216843991789513, 9.433806746869016, 9.920530350465249, 10.22582854171676, 10.77962201040569, 11.48432103998255, 12.24421142845438, 12.64638130283728, 13.03608772973673, 13.43353773926105
