L(s) = 1 | + (0.826 + 0.563i)4-s + (0.955 + 0.294i)7-s + (−0.914 + 0.848i)13-s + (0.365 + 0.930i)16-s − 0.445·19-s + (−0.733 − 0.680i)25-s + (0.623 + 0.781i)28-s + (0.900 + 1.56i)31-s + (1.62 + 0.781i)37-s + (−0.162 − 0.414i)43-s + (0.826 + 0.563i)49-s + (−1.23 + 0.185i)52-s + (−1.48 + 1.01i)61-s + (−0.222 + 0.974i)64-s + (−0.623 − 1.07i)67-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)4-s + (0.955 + 0.294i)7-s + (−0.914 + 0.848i)13-s + (0.365 + 0.930i)16-s − 0.445·19-s + (−0.733 − 0.680i)25-s + (0.623 + 0.781i)28-s + (0.900 + 1.56i)31-s + (1.62 + 0.781i)37-s + (−0.162 − 0.414i)43-s + (0.826 + 0.563i)49-s + (−1.23 + 0.185i)52-s + (−1.48 + 1.01i)61-s + (−0.222 + 0.974i)64-s + (−0.623 − 1.07i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.680383711\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.680383711\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.955 - 0.294i)T \) |
good | 2 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 5 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (0.914 - 0.848i)T + (0.0747 - 0.997i)T^{2} \) |
| 17 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 31 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 41 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 43 | \( 1 + (0.162 + 0.414i)T + (-0.733 + 0.680i)T^{2} \) |
| 47 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 53 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 61 | \( 1 + (1.48 - 1.01i)T + (0.365 - 0.930i)T^{2} \) |
| 67 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 89 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 97 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.610712940590379935381470702537, −7.931882049248359349699389692348, −7.42257691253784613452633359571, −6.56193826719560591624652370859, −5.99659032540875109196718762082, −4.80047368457255530789184377633, −4.37049643726258671880468750489, −3.15837778267057541911604987499, −2.32826139432958307586346591691, −1.59580956207484026405939527494,
0.949386490301175471173429566257, 2.09918340397713015113688968948, 2.73632884031631028325568893189, 4.01272057745953146598971549862, 4.83855260639899136427776090735, 5.60507956240366854147231468009, 6.20069105832742116965232851931, 7.19207229406722334037335730094, 7.74211281637453408004033229149, 8.217828924927580764271813595264