| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 2·13-s + 2·15-s − 17-s − 6·19-s − 21-s + 6·23-s − 25-s − 27-s + 29-s + 4·31-s − 2·35-s + 2·39-s + 9·41-s + 4·43-s − 2·45-s − 3·47-s − 6·49-s + 51-s + 11·53-s + 6·57-s + 11·59-s − 2·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.242·17-s − 1.37·19-s − 0.218·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.718·31-s − 0.338·35-s + 0.320·39-s + 1.40·41-s + 0.609·43-s − 0.298·45-s − 0.437·47-s − 6/7·49-s + 0.140·51-s + 1.51·53-s + 0.794·57-s + 1.43·59-s − 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455755198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455755198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.72871650893717, −13.11262878949804, −12.72297475959249, −12.30328595651627, −11.64820117727266, −11.47304806169751, −10.80747963830269, −10.53949386548587, −9.927217701381040, −9.169008987735823, −8.889173863370789, −8.005621291861192, −7.918637876049768, −7.189568700394597, −6.674748780136129, −6.257343570175275, −5.516554354066375, −4.903478857185921, −4.534647892198133, −3.976086436150848, −3.413682610215195, −2.485754554905768, −2.090560341254759, −0.9956885810710240, −0.4754696705040496,
0.4754696705040496, 0.9956885810710240, 2.090560341254759, 2.485754554905768, 3.413682610215195, 3.976086436150848, 4.534647892198133, 4.903478857185921, 5.516554354066375, 6.257343570175275, 6.674748780136129, 7.189568700394597, 7.918637876049768, 8.005621291861192, 8.889173863370789, 9.169008987735823, 9.927217701381040, 10.53949386548587, 10.80747963830269, 11.47304806169751, 11.64820117727266, 12.30328595651627, 12.72297475959249, 13.11262878949804, 13.72871650893717
