L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s − 0.999·12-s + (−0.5 + 0.866i)16-s − 1.73i·17-s + 0.999·18-s + (0.5 + 0.866i)19-s + (1.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (0.5 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s − 0.999·12-s + (−0.5 + 0.866i)16-s − 1.73i·17-s + 0.999·18-s + (0.5 + 0.866i)19-s + (1.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7576253594\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7576253594\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.432458904204867920238308335262, −8.501454832972385150934496193076, −8.019836385334356519837661368836, −7.33176718438473928179658605471, −6.56436377824123835543205572144, −5.62438940951620278398759517304, −4.95198298515001105788472952546, −3.34901260044235967539600621043, −2.29550226408341643665090498388, −0.69422440525209634359608093189,
1.89987665586571166403869165124, 2.84177364312306092326679357149, 3.71961002244261378867172309241, 4.71907367674055481386319982272, 5.36404037193154562530180335149, 6.99268630818163079759017963548, 7.978537459047402292229944809644, 8.379189087072538668641639368309, 9.395154639966438386803566134467, 9.914239862927693526390715025884