L(s) = 1 | + (−0.0327 − 0.999i)3-s + (0.352 + 0.935i)5-s + (−0.997 + 0.0654i)9-s + (−0.227 − 0.973i)11-s + (−0.773 + 0.634i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (0.812 + 0.582i)19-s + (−0.659 + 0.751i)23-s + (−0.751 + 0.659i)25-s + (0.0980 + 0.995i)27-s + (−0.290 − 0.956i)29-s + (−0.965 − 0.258i)31-s + (−0.965 + 0.258i)33-s + (0.412 − 0.910i)37-s + ⋯ |
L(s) = 1 | + (−0.0327 − 0.999i)3-s + (0.352 + 0.935i)5-s + (−0.997 + 0.0654i)9-s + (−0.227 − 0.973i)11-s + (−0.773 + 0.634i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (0.812 + 0.582i)19-s + (−0.659 + 0.751i)23-s + (−0.751 + 0.659i)25-s + (0.0980 + 0.995i)27-s + (−0.290 − 0.956i)29-s + (−0.965 − 0.258i)31-s + (−0.965 + 0.258i)33-s + (0.412 − 0.910i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1905626987 + 0.3216638601i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1905626987 + 0.3216638601i\) |
\(L(1)\) |
\(\approx\) |
\(0.8235000488 - 0.09772618557i\) |
\(L(1)\) |
\(\approx\) |
\(0.8235000488 - 0.09772618557i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.0327 - 0.999i)T \) |
| 5 | \( 1 + (0.352 + 0.935i)T \) |
| 11 | \( 1 + (-0.227 - 0.973i)T \) |
| 13 | \( 1 + (-0.773 + 0.634i)T \) |
| 17 | \( 1 + (-0.130 - 0.991i)T \) |
| 19 | \( 1 + (0.812 + 0.582i)T \) |
| 23 | \( 1 + (-0.659 + 0.751i)T \) |
| 29 | \( 1 + (-0.290 - 0.956i)T \) |
| 31 | \( 1 + (-0.965 - 0.258i)T \) |
| 37 | \( 1 + (0.412 - 0.910i)T \) |
| 41 | \( 1 + (-0.195 + 0.980i)T \) |
| 43 | \( 1 + (-0.881 + 0.471i)T \) |
| 47 | \( 1 + (0.608 + 0.793i)T \) |
| 53 | \( 1 + (0.973 - 0.227i)T \) |
| 59 | \( 1 + (-0.162 + 0.986i)T \) |
| 61 | \( 1 + (-0.528 - 0.849i)T \) |
| 67 | \( 1 + (-0.999 + 0.0327i)T \) |
| 71 | \( 1 + (-0.555 + 0.831i)T \) |
| 73 | \( 1 + (0.896 + 0.442i)T \) |
| 79 | \( 1 + (0.991 + 0.130i)T \) |
| 83 | \( 1 + (-0.995 - 0.0980i)T \) |
| 89 | \( 1 + (-0.321 + 0.946i)T \) |
| 97 | \( 1 + (-0.707 + 0.707i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.21175738718764304162618156802, −19.6559059539075849756951784560, −18.21080420220674626493985827896, −17.65051182607773900368429187140, −16.85099338399267148277343406659, −16.44210078600436841693520216534, −15.38978863912103276696073698803, −15.05684629968069975460397578705, −14.1192166557872910723395576250, −13.2261994075132171085604501134, −12.40407312232203640641599259198, −11.90258913220667046739753718589, −10.62445622813959236488804515123, −10.192469650372181358667251427064, −9.41484473126277128678985108541, −8.75992848713913312399740617647, −7.96215407988098372877322228128, −6.956942029309130741490961245054, −5.74585826759863444116848058909, −5.15994691942104852746254927145, −4.544531544673607230576350188522, −3.676666808402218306052424995559, −2.58608308928750183787681111139, −1.64374240183287901575584339546, −0.12631314372783956267358693010,
1.30629966125608347907155700515, 2.3197878381828918757718402472, 2.908660270026299644176296225324, 3.88233322428470230391619651090, 5.35466527718710093888357152519, 5.88783136162768987771543296024, 6.73158486961359918519882950900, 7.48802454446437153673248227985, 7.95880830808599758139344090953, 9.22691581227027675305209487392, 9.78036712102776900889470595776, 10.940721262043920490176661348841, 11.53034194291330864415275938395, 12.08178523046684066935784323227, 13.25229575485712318348901079900, 13.77452337561160160994600121723, 14.29276747153994837801926428702, 15.062858460929863077753706424240, 16.21832443191606974880735737971, 16.79965278030590586594353960561, 17.81110976986926896854662216470, 18.27617803102239624347005409982, 18.85824802570464878088858633692, 19.526621376216098397911758025502, 20.25846852500016814227573930539