L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 14-s + 16-s − 4·17-s − 6·19-s − 20-s + 25-s − 28-s − 8·31-s + 32-s − 4·34-s + 35-s + 6·37-s − 6·38-s − 40-s − 6·41-s + 8·43-s − 12·47-s + 49-s + 50-s − 6·53-s − 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.37·19-s − 0.223·20-s + 1/5·25-s − 0.188·28-s − 1.43·31-s + 0.176·32-s − 0.685·34-s + 0.169·35-s + 0.986·37-s − 0.973·38-s − 0.158·40-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.141·50-s − 0.824·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.92807828398354, −12.67069047174188, −12.32010051821985, −11.68535794781981, −11.19874604501877, −10.86540358385100, −10.61827055172671, −9.812955602523174, −9.436930065436532, −8.934158923245353, −8.368548285294291, −7.990124945097192, −7.441175738804772, −6.806969695448628, −6.608449747118250, −6.056709602931868, −5.559604619468508, −4.935301938796349, −4.442503554898902, −4.057897099654108, −3.592089416620302, −2.912233180460881, −2.499407169100284, −1.817700638923464, −1.283779347256611, 0, 0,
1.283779347256611, 1.817700638923464, 2.499407169100284, 2.912233180460881, 3.592089416620302, 4.057897099654108, 4.442503554898902, 4.935301938796349, 5.559604619468508, 6.056709602931868, 6.608449747118250, 6.806969695448628, 7.441175738804772, 7.990124945097192, 8.368548285294291, 8.934158923245353, 9.436930065436532, 9.812955602523174, 10.61827055172671, 10.86540358385100, 11.19874604501877, 11.68535794781981, 12.32010051821985, 12.67069047174188, 12.92807828398354