L(s) = 1 | + (−1.25 − 0.656i)2-s + (−1.95 + 0.523i)3-s + (1.13 + 1.64i)4-s + (−0.959 − 0.256i)5-s + (2.78 + 0.625i)6-s + (−0.348 − 2.80i)8-s + (0.941 − 0.543i)9-s + (1.03 + 0.951i)10-s + (−0.505 − 1.88i)11-s + (−3.08 − 2.61i)12-s + (2.10 + 2.10i)13-s + 2.00·15-s + (−1.40 + 3.74i)16-s + (−2.83 + 4.91i)17-s + (−1.53 + 0.0632i)18-s + (0.165 − 0.616i)19-s + ⋯ |
L(s) = 1 | + (−0.885 − 0.463i)2-s + (−1.12 + 0.302i)3-s + (0.569 + 0.821i)4-s + (−0.428 − 0.114i)5-s + (1.13 + 0.255i)6-s + (−0.123 − 0.992i)8-s + (0.313 − 0.181i)9-s + (0.326 + 0.300i)10-s + (−0.152 − 0.569i)11-s + (−0.890 − 0.754i)12-s + (0.583 + 0.583i)13-s + 0.518·15-s + (−0.351 + 0.936i)16-s + (−0.688 + 1.19i)17-s + (−0.362 + 0.0149i)18-s + (0.0379 − 0.141i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.287217 - 0.244451i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.287217 - 0.244451i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.25 + 0.656i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.95 - 0.523i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (0.959 + 0.256i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (0.505 + 1.88i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.10 - 2.10i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.83 - 4.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.165 + 0.616i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (5.92 - 3.42i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.207 + 0.207i)T + 29iT^{2} \) |
| 31 | \( 1 + (-3.94 + 6.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.60 - 2.57i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.40iT - 41T^{2} \) |
| 43 | \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.144 - 0.250i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.04 + 7.62i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.60 + 13.4i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.969 + 3.61i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-10.0 + 2.69i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (-0.310 - 0.179i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.84 + 6.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.424 - 0.424i)T + 83iT^{2} \) |
| 89 | \( 1 + (-15.2 + 8.79i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−10.19878761623853973364232361739, −9.457762355073792495188129054453, −8.311105719395842233491155130208, −7.88093780119573664792760770584, −6.37445805533469685002371972568, −6.03968463671814746127371185041, −4.47293900215290904357547308404, −3.66860048958708332795281766869, −2.04001605171245358841359527954, −0.38633929440151112165487376897,
0.935813266421926753150845721841, 2.59866831683274430822163218038, 4.42303474783581914908085256826, 5.50703514196591534576044558487, 6.21187171216498457996711793226, 7.04736623084997467708501294055, 7.78283411210433040542623369427, 8.705215513655545171696114903779, 9.695284786738076283670297113931, 10.53617562651883200377241418063