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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4197590113\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4197590113\) |
\(L(1)\) |
|
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
−9.895043835989397909750700905571, −9.249109080666133669029647476216, −8.170905807196320152756144609928, −7.73231618671429380129992491498, −6.42291366203138104617980211209, −5.90881861958821560576394813927, −4.95957710540182846203141573479, −4.17548716303762054826363088153, −2.02682068600729289286715810061, −0.53759207857269750250634367840,
1.85365109439781258637737485083, 2.81676272422082335087738547439, 3.86277527193516566587152088259, 5.51191391473355030628574538169, 6.05555187545371321061648656789, 7.12395030998808269427254578176, 7.944309890380718991402839368838, 9.003608074628267204956649356531, 9.594237872556942264247598357773, 10.50460638259620096505839778185