L(s) = 1 | + (0.733 − 0.680i)2-s + (0.722 − 0.108i)3-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (0.455 − 0.571i)6-s + (−0.0747 + 0.997i)7-s + (−0.623 − 0.781i)8-s + (−0.445 + 0.137i)9-s + (−0.365 − 0.930i)10-s + (−0.0546 − 0.728i)12-s + (0.623 + 0.781i)14-s + (0.162 − 0.712i)15-s + (−0.988 − 0.149i)16-s + (−0.233 + 0.403i)18-s + (−0.900 − 0.433i)20-s + (0.0546 + 0.728i)21-s + ⋯ |
L(s) = 1 | + (0.733 − 0.680i)2-s + (0.722 − 0.108i)3-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (0.455 − 0.571i)6-s + (−0.0747 + 0.997i)7-s + (−0.623 − 0.781i)8-s + (−0.445 + 0.137i)9-s + (−0.365 − 0.930i)10-s + (−0.0546 − 0.728i)12-s + (0.623 + 0.781i)14-s + (0.162 − 0.712i)15-s + (−0.988 − 0.149i)16-s + (−0.233 + 0.403i)18-s + (−0.900 − 0.433i)20-s + (0.0546 + 0.728i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.795965383\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.795965383\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.733 + 0.680i)T \) |
| 5 | \( 1 + (-0.365 + 0.930i)T \) |
| 7 | \( 1 + (0.0747 - 0.997i)T \) |
good | 3 | \( 1 + (-0.722 + 0.108i)T + (0.955 - 0.294i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.63 - 1.11i)T + (0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 41 | \( 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (1.19 - 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.914 + 0.848i)T + (0.0747 - 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (0.147 + 1.97i)T + (-0.988 + 0.149i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.0332 + 0.145i)T + (-0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.706691113412387146508813059997, −9.326941673125162177977065380918, −8.637591471516904571430841493530, −7.64748466465604297004907222700, −6.21334587773823504710467518829, −5.48847639823711657946337207096, −4.80139838931463354868514016871, −3.48992051266533769164603487099, −2.59006719863379730046151873032, −1.60561369764334737488528690094,
2.39648919328569146853149944006, 3.30458613928910053006148989174, 4.00047323969555791638395892423, 5.26021309395698575867664900341, 6.20986884731605788703622947891, 7.13658811872638273567288839503, 7.51252076150191815476583830202, 8.710919163387870846996683433529, 9.296092267254613746545092558582, 10.59255302300550425666497443340