L(s) = 1 | + (0.568 + 0.822i)2-s + (−0.748 − 0.663i)3-s + (−0.354 + 0.935i)4-s + (0.120 + 0.992i)5-s + (0.120 − 0.992i)6-s + (0.568 + 0.822i)7-s + (−0.970 + 0.239i)8-s + (0.120 + 0.992i)9-s + (−0.748 + 0.663i)10-s + (0.885 − 0.464i)11-s + (0.885 − 0.464i)12-s + (−0.354 − 0.935i)13-s + (−0.354 + 0.935i)14-s + (0.568 − 0.822i)15-s + (−0.748 − 0.663i)16-s + (−0.970 − 0.239i)17-s + ⋯ |
L(s) = 1 | + (0.568 + 0.822i)2-s + (−0.748 − 0.663i)3-s + (−0.354 + 0.935i)4-s + (0.120 + 0.992i)5-s + (0.120 − 0.992i)6-s + (0.568 + 0.822i)7-s + (−0.970 + 0.239i)8-s + (0.120 + 0.992i)9-s + (−0.748 + 0.663i)10-s + (0.885 − 0.464i)11-s + (0.885 − 0.464i)12-s + (−0.354 − 0.935i)13-s + (−0.354 + 0.935i)14-s + (0.568 − 0.822i)15-s + (−0.748 − 0.663i)16-s + (−0.970 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6840620450 + 0.6251438268i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6840620450 + 0.6251438268i\) |
\(L(1)\) |
\(\approx\) |
\(0.9298485393 + 0.5149197180i\) |
\(L(1)\) |
\(\approx\) |
\(0.9298485393 + 0.5149197180i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (0.568 + 0.822i)T \) |
| 3 | \( 1 + (-0.748 - 0.663i)T \) |
| 5 | \( 1 + (0.120 + 0.992i)T \) |
| 7 | \( 1 + (0.568 + 0.822i)T \) |
| 11 | \( 1 + (0.885 - 0.464i)T \) |
| 13 | \( 1 + (-0.354 - 0.935i)T \) |
| 17 | \( 1 + (-0.970 - 0.239i)T \) |
| 19 | \( 1 + (-0.354 - 0.935i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.885 + 0.464i)T \) |
| 31 | \( 1 + (0.885 + 0.464i)T \) |
| 37 | \( 1 + (-0.748 - 0.663i)T \) |
| 41 | \( 1 + (0.885 - 0.464i)T \) |
| 43 | \( 1 + (-0.748 + 0.663i)T \) |
| 47 | \( 1 + (0.120 - 0.992i)T \) |
| 59 | \( 1 + (0.120 - 0.992i)T \) |
| 61 | \( 1 + (-0.970 + 0.239i)T \) |
| 67 | \( 1 + (-0.354 + 0.935i)T \) |
| 71 | \( 1 + (-0.748 + 0.663i)T \) |
| 73 | \( 1 + (-0.970 - 0.239i)T \) |
| 79 | \( 1 + (0.568 - 0.822i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.970 - 0.239i)T \) |
| 97 | \( 1 + (0.120 + 0.992i)T \) |
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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
−33.113163127116821660091624358647, −32.04477776110681915680584503319, −30.83679626777601749151938699240, −29.4611127601006145857392928140, −28.68246773916791553107839101777, −27.648979445921168212621597039535, −26.8407928289077100852901229271, −24.58742858978730033351256179297, −23.62721126462385551474375816218, −22.623585439226291833883195805462, −21.32951023870675571290875824647, −20.65833563763901067674154823485, −19.49653614707637917973953735521, −17.5752863411013350802647764977, −16.74749055145409988478799859121, −15.124537989161103948586988458689, −13.81462977540892684746508877218, −12.34254450265334822916515601110, −11.43669352036156958526524761721, −10.13666626876480964380864644280, −8.99840237038707639287607138684, −6.405887794373231569034573778711, −4.70810027039232146783674162679, −4.190830438108280453535043392187, −1.41049037409665203179351974939,
2.715021785663709957518028222806, 4.95036073895972449844555021601, 6.22256016249025029841318506690, 7.1492026754325983208232384759, 8.6616993347269998152844996632, 11.00001396138994479506741125315, 12.046190476969748770819900088793, 13.40238529785362593371472326562, 14.63022209630754673626045078066, 15.713675268556297948590634065982, 17.42902427756675388888030490002, 17.91242929044823670662637718304, 19.29168452677879223262602227057, 21.61594633230051699966079419184, 22.26385085009380096919582637900, 23.22939864849284858936967186781, 24.61035010893629838022049675963, 25.09244323343132838033464101381, 26.7153770056248746514368530834, 27.781655994174021581501576236052, 29.49432483336252392996623621434, 30.366061887097786236050809782812, 31.18245821052252693661915054149, 32.76766790840647752892317306900, 33.80712852206733400869168170412