L(s) = 1 | + (0.994 − 0.104i)2-s + (0.743 + 0.669i)3-s + (0.978 − 0.207i)4-s + (0.809 + 0.587i)6-s + (0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (0.866 + 0.5i)12-s + (0.587 + 0.809i)13-s + (0.913 − 0.406i)16-s + (−0.994 − 0.104i)17-s + (0.207 + 0.978i)18-s + (−0.978 − 0.207i)19-s + (−0.866 − 0.5i)23-s + (0.913 + 0.406i)24-s + (0.669 + 0.743i)26-s + (−0.587 + 0.809i)27-s + ⋯ |
L(s) = 1 | + (0.994 − 0.104i)2-s + (0.743 + 0.669i)3-s + (0.978 − 0.207i)4-s + (0.809 + 0.587i)6-s + (0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (0.866 + 0.5i)12-s + (0.587 + 0.809i)13-s + (0.913 − 0.406i)16-s + (−0.994 − 0.104i)17-s + (0.207 + 0.978i)18-s + (−0.978 − 0.207i)19-s + (−0.866 − 0.5i)23-s + (0.913 + 0.406i)24-s + (0.669 + 0.743i)26-s + (−0.587 + 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.970062640 + 0.8491581760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.970062640 + 0.8491581760i\) |
\(L(1)\) |
\(\approx\) |
\(2.284636438 + 0.3942574383i\) |
\(L(1)\) |
\(\approx\) |
\(2.284636438 + 0.3942574383i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.994 - 0.104i)T \) |
| 3 | \( 1 + (0.743 + 0.669i)T \) |
| 13 | \( 1 + (0.587 + 0.809i)T \) |
| 17 | \( 1 + (-0.994 - 0.104i)T \) |
| 19 | \( 1 + (-0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.913 + 0.406i)T \) |
| 37 | \( 1 + (0.743 - 0.669i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.207 + 0.978i)T \) |
| 53 | \( 1 + (-0.406 + 0.913i)T \) |
| 59 | \( 1 + (0.978 - 0.207i)T \) |
| 61 | \( 1 + (-0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.207 + 0.978i)T \) |
| 79 | \( 1 + (-0.104 - 0.994i)T \) |
| 83 | \( 1 + (-0.587 + 0.809i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.587 + 0.809i)T \) |
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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
−24.334839240582385262746042522560, −23.64100131065849328301467081729, −22.86115260788458383110722347963, −21.803919823798632688959896657108, −20.95517254287752635999647432843, −20.06560288993220611219295808681, −19.56819296667595687157176844652, −18.322893761297448419693527092809, −17.45031124193326969720022238133, −16.177890537233854030725804911991, −15.23653623094080262262974493858, −14.64486962210448213227909911696, −13.49029354972738114716849317592, −13.121885923963710522556825969, −12.14386816896655461696488028341, −11.162531682007482416714031419933, −10.01354483632861356836362970480, −8.51697405936321660796627133452, −7.876483055447737836220179624270, −6.66038241064436737561313749985, −6.04473057562892683256852377485, −4.595183212213505293901961819977, −3.54817990681987863666149794949, −2.59036707090640778534544439280, −1.501159683459285118785910920322,
1.9176296928734645415990358824, 2.75389124564427136127511340742, 4.13479651628793491640543068068, 4.43576630674686042511740183138, 5.895964021403638959766132684578, 6.86711244512036971940777267331, 8.131719256991610763542820997473, 9.09806583572666388730928873688, 10.3081634807280018793480519679, 11.03568225287043383073455056025, 12.08663170021914161484233322777, 13.2900225852637448004646819090, 13.8747636976110620099569720588, 14.727687074712761889835759829199, 15.630808486394538229385145910748, 16.169464951553588368654318354692, 17.30118303863195816387756874768, 18.87014408527979076674075541236, 19.59287979119276851792265328213, 20.475298463698831627121958498253, 21.166807490308779207252316453573, 21.870299623970169323668255936402, 22.67137416040247713116842818172, 23.70815330040670159822778531314, 24.515403999002041729587798904548