L(s) = 1 | + (2.42 + 1.76i)3-s + (1.61 − 1.17i)4-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (1.85 + 5.70i)9-s + 6·12-s + (1.23 + 3.80i)13-s + (−2.42 + 1.76i)15-s + (1.23 − 3.80i)16-s + (−0.618 + 1.90i)17-s + (−4.85 − 3.52i)19-s + (0.618 + 1.90i)20-s − 3·21-s − 5·23-s + (3.23 + 2.35i)25-s + ⋯ |
L(s) = 1 | + (1.40 + 1.01i)3-s + (0.809 − 0.587i)4-s + (−0.138 + 0.425i)5-s + (−0.305 + 0.222i)7-s + (0.618 + 1.90i)9-s + 1.73·12-s + (0.342 + 1.05i)13-s + (−0.626 + 0.455i)15-s + (0.309 − 0.951i)16-s + (−0.149 + 0.461i)17-s + (−1.11 − 0.809i)19-s + (0.138 + 0.425i)20-s − 0.654·21-s − 1.04·23-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
\(L(1)\) |
\(\approx\) |
\(2.37170 + 1.48292i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.37170 + 1.48292i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.42 - 1.76i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (0.309 - 0.951i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 3.80i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.618 - 1.90i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (4.85 + 3.52i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 + (-8.09 + 5.87i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.04 + 2.93i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (1.61 + 1.17i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (6.47 + 4.70i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.85 + 5.70i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.42 - 1.76i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.618 + 1.90i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.09 + 5.87i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.85 + 5.70i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.70 - 11.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + (1.54 + 4.75i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.24938310000116205102738264675, −9.540187014334091998859021516652, −8.777469420663855341100451389518, −7.995455202061340822625152718145, −6.86080151981069932241731810699, −6.16283522315935989065843124132, −4.71158784166738734291601555283, −3.87159897017269670202448671487, −2.75878029288106046887440356866, −2.05131082770292051708494333216,
1.27831561270769580785015426772, 2.53766212771330727118091720749, 3.20103526466802375520514475617, 4.27340117459145478232609343687, 6.10074884962540342783625618715, 6.76930020858413221706683836030, 7.71783801418545149448320001522, 8.226540829701649048991908317678, 8.752534845451673185531178972594, 9.970960891042051973000426335792