L(s) = 1 | + (−0.809 − 0.587i)3-s + (1.61 − 1.17i)4-s + (0.927 − 2.85i)5-s + (0.809 − 0.587i)7-s + (−0.618 − 1.90i)9-s − 2·12-s + (1.23 + 3.80i)13-s + (−2.42 + 1.76i)15-s + (1.23 − 3.80i)16-s + (1.85 − 5.70i)17-s + (1.61 + 1.17i)19-s + (−1.85 − 5.70i)20-s − 21-s + 3·23-s + (−3.23 − 2.35i)25-s + ⋯ |
L(s) = 1 | + (−0.467 − 0.339i)3-s + (0.809 − 0.587i)4-s + (0.414 − 1.27i)5-s + (0.305 − 0.222i)7-s + (−0.206 − 0.634i)9-s − 0.577·12-s + (0.342 + 1.05i)13-s + (−0.626 + 0.455i)15-s + (0.309 − 0.951i)16-s + (0.449 − 1.38i)17-s + (0.371 + 0.269i)19-s + (−0.414 − 1.27i)20-s − 0.218·21-s + 0.625·23-s + (−0.647 − 0.470i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.927402 - 1.48323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.927402 - 1.48323i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.809 + 0.587i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.927 + 2.85i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 3.80i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.85 + 5.70i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.61 - 1.17i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + (4.85 - 3.52i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.54 - 4.75i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (8.89 - 6.46i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.85 - 3.52i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.85 + 5.70i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.28 + 5.29i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.09 + 9.51i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + (-2.78 + 8.55i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.61 + 1.17i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.09 - 9.51i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.70 - 11.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (0.309 + 0.951i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.722944383243297767037682785596, −9.278430195129973166812183551724, −8.285759716846356483852797913210, −7.00829409040154484338319263673, −6.56987027652100956819351416758, −5.29634439035559698864153169438, −5.05204443259335766076995083187, −3.37373394406781363109692348136, −1.72017520846027672084222802139, −0.943759179265740475226094047146,
2.02142050438490534403296989004, 2.94369367897533374593489934246, 3.91302068954960199188422383879, 5.54628317661524324980862582916, 5.97566144810995887583867091023, 7.07137314562733804446589328904, 7.77672820547820732293884773351, 8.591437273370109969760579453360, 10.06370144984573970841377826030, 10.61778508352862002501737589978