L(s) = 1 | + (0.733 + 0.680i)4-s + (0.988 − 0.149i)7-s + (1.85 + 0.728i)13-s + (0.0747 + 0.997i)16-s + (−1.35 − 0.781i)19-s + (0.623 − 0.781i)25-s + (0.826 + 0.563i)28-s + (0.510 + 0.294i)31-s + (−1.82 − 0.563i)37-s + (−0.603 + 0.411i)43-s + (0.955 − 0.294i)49-s + (0.865 + 1.79i)52-s + (−0.925 − 0.997i)61-s + (−0.623 + 0.781i)64-s + (−0.826 + 1.43i)67-s + ⋯ |
L(s) = 1 | + (0.733 + 0.680i)4-s + (0.988 − 0.149i)7-s + (1.85 + 0.728i)13-s + (0.0747 + 0.997i)16-s + (−1.35 − 0.781i)19-s + (0.623 − 0.781i)25-s + (0.826 + 0.563i)28-s + (0.510 + 0.294i)31-s + (−1.82 − 0.563i)37-s + (−0.603 + 0.411i)43-s + (0.955 − 0.294i)49-s + (0.865 + 1.79i)52-s + (−0.925 − 0.997i)61-s + (−0.623 + 0.781i)64-s + (−0.826 + 1.43i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.859378536\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.859378536\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.988 + 0.149i)T \) |
good | 2 | \( 1 + (-0.733 - 0.680i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 13 | \( 1 + (-1.85 - 0.728i)T + (0.733 + 0.680i)T^{2} \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 19 | \( 1 + (1.35 + 0.781i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (0.826 + 0.563i)T^{2} \) |
| 31 | \( 1 + (-0.510 - 0.294i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.82 + 0.563i)T + (0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 43 | \( 1 + (0.603 - 0.411i)T + (0.365 - 0.930i)T^{2} \) |
| 47 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 53 | \( 1 + (0.826 - 0.563i)T^{2} \) |
| 59 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 61 | \( 1 + (0.925 + 0.997i)T + (-0.0747 + 0.997i)T^{2} \) |
| 67 | \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.290 + 1.92i)T + (-0.955 - 0.294i)T^{2} \) |
| 79 | \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 89 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 + (1.61 + 0.930i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.468167114327288638640918418917, −8.223697676122238876360050496089, −7.11156373276405799972546675655, −6.61573922840521827387328455372, −5.94647030440384331094541409226, −4.76330606465897077757106883103, −4.11025628631761069561007679943, −3.29687769874291435967243641306, −2.19848002221238223474653054591, −1.43651905514544621111065399326,
1.25692908219981936707296279245, 1.86287207736807130923996757657, 3.04830946992224084923453153917, 3.95505862482543909024338425043, 4.99375276443798698008359809576, 5.67828167087610620012517400352, 6.28276537102564338457783301024, 6.99679148762186946140163700081, 7.989942808595970988040222748034, 8.456215557528429753543757027995