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
−10.53617562651883200377241418063, −9.695284786738076283670297113931, −8.705215513655545171696114903779, −7.78283411210433040542623369427, −7.04736623084997467708501294055, −6.21187171216498457996711793226, −5.50703514196591534576044558487, −4.42303474783581914908085256826, −2.59866831683274430822163218038, −0.935813266421926753150845721841,
0.38633929440151112165487376897, 2.04001605171245358841359527954, 3.66860048958708332795281766869, 4.47293900215290904357547308404, 6.03968463671814746127371185041, 6.37445805533469685002371972568, 7.88093780119573664792760770584, 8.311105719395842233491155130208, 9.457762355073792495188129054453, 10.19878761623853973364232361739