L(s) = 1 | − 1.41·2-s + 2.80i·3-s + 2.00·4-s − 2.23i·5-s − 3.96i·6-s − 2.82·8-s − 4.87·9-s + 3.16i·10-s + 5.61i·12-s + 6.27·15-s + 4.00·16-s + 6.89·18-s − 4.47i·20-s − 8.92·23-s − 7.93i·24-s − 5.00·25-s + ⋯ |
L(s) = 1 | − 1.00·2-s + 1.61i·3-s + 1.00·4-s − 0.999i·5-s − 1.61i·6-s − 1.00·8-s − 1.62·9-s + 1.00i·10-s + 1.61i·12-s + 1.61·15-s + 1.00·16-s + 1.62·18-s − 1.00i·20-s − 1.86·23-s − 1.61i·24-s − 1.00·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0188019 - 0.0411555i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0188019 - 0.0411555i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.80iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8.92T + 23T^{2} \) |
| 29 | \( 1 + 10.7T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 8.15iT - 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 + 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 0.219iT - 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 8.72iT - 83T^{2} \) |
| 89 | \( 1 + 3.74iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.06767900228253195341492919857, −9.815947751302136050543101945765, −8.999336299652403798181640910921, −8.396752480176664735117686816100, −7.54043861339985546885288821711, −6.06547347316021566734424467456, −5.36898680082677842806823939039, −4.28877820253364031855609436374, −3.44165664745150013588051291721, −1.89488625303067155853882298580,
0.02673569747908123961108830157, 1.70480636029277950604414073097, 2.39860744641430917434239684386, 3.59429385993523259345075704428, 5.84842526130417776428142644500, 6.25102055850629639123045426363, 7.30085471227127875471953878650, 7.54740090156811550895084319566, 8.389283322815348124633758581029, 9.414043821241683648951137000421