L(s) = 1 | + (0.401 − 0.915i)3-s + (0.789 − 0.614i)5-s − 7-s + (−0.677 − 0.735i)9-s + (0.879 − 0.475i)11-s + (−0.401 − 0.915i)13-s + (−0.245 − 0.969i)15-s + (−0.986 + 0.164i)17-s + (−0.546 − 0.837i)19-s + (−0.401 + 0.915i)21-s + (−0.546 − 0.837i)23-s + (0.245 − 0.969i)25-s + (−0.945 + 0.324i)27-s + (0.245 + 0.969i)29-s + (0.677 + 0.735i)31-s + ⋯ |
L(s) = 1 | + (0.401 − 0.915i)3-s + (0.789 − 0.614i)5-s − 7-s + (−0.677 − 0.735i)9-s + (0.879 − 0.475i)11-s + (−0.401 − 0.915i)13-s + (−0.245 − 0.969i)15-s + (−0.986 + 0.164i)17-s + (−0.546 − 0.837i)19-s + (−0.401 + 0.915i)21-s + (−0.546 − 0.837i)23-s + (0.245 − 0.969i)25-s + (−0.945 + 0.324i)27-s + (0.245 + 0.969i)29-s + (0.677 + 0.735i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4538716254 - 0.9217324137i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4538716254 - 0.9217324137i\) |
\(L(1)\) |
\(\approx\) |
\(0.8518860501 - 0.6153477240i\) |
\(L(1)\) |
\(\approx\) |
\(0.8518860501 - 0.6153477240i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 191 | \( 1 \) |
good | 3 | \( 1 + (0.401 - 0.915i)T \) |
| 5 | \( 1 + (0.789 - 0.614i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.879 - 0.475i)T \) |
| 13 | \( 1 + (-0.401 - 0.915i)T \) |
| 17 | \( 1 + (-0.986 + 0.164i)T \) |
| 19 | \( 1 + (-0.546 - 0.837i)T \) |
| 23 | \( 1 + (-0.546 - 0.837i)T \) |
| 29 | \( 1 + (0.245 + 0.969i)T \) |
| 31 | \( 1 + (0.677 + 0.735i)T \) |
| 37 | \( 1 + (-0.677 + 0.735i)T \) |
| 41 | \( 1 + (0.945 + 0.324i)T \) |
| 43 | \( 1 + (0.0825 - 0.996i)T \) |
| 47 | \( 1 + (0.879 - 0.475i)T \) |
| 53 | \( 1 + (-0.879 + 0.475i)T \) |
| 59 | \( 1 + (0.677 + 0.735i)T \) |
| 61 | \( 1 + (-0.986 - 0.164i)T \) |
| 67 | \( 1 + (0.986 + 0.164i)T \) |
| 71 | \( 1 + (-0.945 - 0.324i)T \) |
| 73 | \( 1 + (-0.879 - 0.475i)T \) |
| 79 | \( 1 + (-0.245 + 0.969i)T \) |
| 83 | \( 1 + (-0.546 + 0.837i)T \) |
| 89 | \( 1 + (-0.0825 - 0.996i)T \) |
| 97 | \( 1 + (-0.677 + 0.735i)T \) |
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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
−22.49152045043627560453603022197, −21.94895721764521531710365300341, −21.24074281575453755098170007665, −20.38458543496169871947588893887, −19.347328624761035473190809958067, −19.07774890140112867166898002142, −17.59722412919455598166757257463, −17.09396995672314892298422899099, −16.13541677935373658200685943838, −15.40693552390747287800676503385, −14.46470283146201366626926383206, −13.96181935497108938040258847319, −13.066509542573545276819765725773, −11.88566931013602357673687503923, −10.976990200530877162876123723356, −9.93493539225246731818010637885, −9.58249178923841492776629358250, −8.8754718190327413598202960268, −7.48433634308214943629141176022, −6.45782810878950382836495306718, −5.8642916507737307262075692701, −4.43373795704037588609067152543, −3.80338623372024093244340915402, −2.64093700405861230084050444895, −1.89442993807719097904929762061,
0.21219226264160080355288724731, 1.14258135807926102711046653675, 2.34399963205928880517600572772, 3.101446031764931382706075290164, 4.407689993308853647050207050813, 5.70767409658566831978920388949, 6.44816286553310964894025631697, 7.05509287261719954908930950297, 8.57795398419768204881226526289, 8.81380715349636324260398101176, 9.8533069362147259113160233619, 10.83024706542748819746367471861, 12.23831612242570141562176521813, 12.60438508625938669461161391073, 13.48108988562396814890312959725, 13.990424884614366865882993206569, 15.0677538504328198827873075078, 16.04790342649814597767207450262, 17.08199447528579585251462929112, 17.54989856361171648083446039494, 18.44852902315973907967399719513, 19.49715116795801013671074285924, 19.86983120376017956782680601171, 20.622468386209711436265342833517, 21.92134769246687750135004566892