L(s) = 1 | + (−0.142 − 0.989i)3-s + (1.21 + 1.16i)7-s + (−0.959 + 0.281i)9-s + (1.42 − 0.273i)13-s + (−0.205 + 0.196i)19-s + (0.975 − 1.37i)21-s + (−0.654 − 0.755i)25-s + (0.415 + 0.909i)27-s + (−1.95 − 0.376i)31-s + (0.654 − 1.13i)37-s + (−0.473 − 1.36i)39-s + (−0.550 − 0.353i)43-s + (0.0871 + 1.82i)49-s + (0.223 + 0.175i)57-s + (1.56 + 0.149i)61-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)3-s + (1.21 + 1.16i)7-s + (−0.959 + 0.281i)9-s + (1.42 − 0.273i)13-s + (−0.205 + 0.196i)19-s + (0.975 − 1.37i)21-s + (−0.654 − 0.755i)25-s + (0.415 + 0.909i)27-s + (−1.95 − 0.376i)31-s + (0.654 − 1.13i)37-s + (−0.473 − 1.36i)39-s + (−0.550 − 0.353i)43-s + (0.0871 + 1.82i)49-s + (0.223 + 0.175i)57-s + (1.56 + 0.149i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.045789959\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.045789959\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (0.327 - 0.945i)T \) |
good | 5 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 7 | \( 1 + (-1.21 - 1.16i)T + (0.0475 + 0.998i)T^{2} \) |
| 11 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 13 | \( 1 + (-1.42 + 0.273i)T + (0.928 - 0.371i)T^{2} \) |
| 17 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 19 | \( 1 + (0.205 - 0.196i)T + (0.0475 - 0.998i)T^{2} \) |
| 23 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1.95 + 0.376i)T + (0.928 + 0.371i)T^{2} \) |
| 37 | \( 1 + (-0.654 + 1.13i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 43 | \( 1 + (0.550 + 0.353i)T + (0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 53 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-1.56 - 0.149i)T + (0.981 + 0.189i)T^{2} \) |
| 71 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 73 | \( 1 + (1.15 + 0.110i)T + (0.981 + 0.189i)T^{2} \) |
| 79 | \( 1 + (0.642 - 1.85i)T + (-0.786 - 0.618i)T^{2} \) |
| 83 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 89 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.76058982120291004441956158311, −9.268263715686660478495297889273, −8.427994619613486396290813862243, −8.055307808844688319425111228598, −6.96749221785038834535100289845, −5.77504781547844487158596933804, −5.52032789238713373489315621676, −3.98044863221021590497162802478, −2.46797091864087138532156815064, −1.54767577562072779371598694652,
1.53755552265911241379666092597, 3.45355542820890897623435586595, 4.16774655310867951341212936337, 5.02411455058047808060180171951, 5.99923676867595261789915870971, 7.16010556964979231748794909792, 8.140401800015385372060743858326, 8.835270956521519846076621609902, 9.801177517499693176604966553885, 10.71951297644070887249509252967