L(s) = 1 | + (0.382 − 0.662i)2-s + (−0.5 − 0.866i)3-s + (0.207 + 0.358i)4-s + (0.923 − 1.60i)5-s − 0.765·6-s + 1.08·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (−0.382 − 0.662i)11-s + (0.207 − 0.358i)12-s − 13-s − 1.84·15-s + (0.207 − 0.358i)16-s + (0.382 + 0.662i)18-s + 0.765·20-s + ⋯ |
L(s) = 1 | + (0.382 − 0.662i)2-s + (−0.5 − 0.866i)3-s + (0.207 + 0.358i)4-s + (0.923 − 1.60i)5-s − 0.765·6-s + 1.08·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (−0.382 − 0.662i)11-s + (0.207 − 0.358i)12-s − 13-s − 1.84·15-s + (0.207 − 0.358i)16-s + (0.382 + 0.662i)18-s + 0.765·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.449917808\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.449917808\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.84T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 0.765T + T^{2} \) |
| 89 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−9.123578083768688105179514188344, −8.142932349533341060397994479852, −7.73643768018726526098922996859, −6.60707245181531177701951266639, −5.70844900249492314077503060587, −5.04949052705041863000786148130, −4.33889744536100124009003410657, −2.80085278337448965096891717605, −2.01993222046230178137897736561, −1.02268019764110511270924881952,
2.05611580767749605967054170809, 2.92375765469683277407709145621, 4.16973415007004942936615968804, 5.13256260481903109531878663629, 5.69622956218373760390498443685, 6.48696293143417147297671861677, 7.01092822703179509628583612825, 7.75200933715683022700033641635, 9.294800571696405405896255204477, 9.950756835600683855745170909590