L(s) = 1 | + (0.458 + 0.888i)2-s + (−0.866 + 0.5i)3-s + (−0.580 + 0.814i)4-s + (−0.841 − 0.540i)6-s + (−0.989 − 0.142i)8-s + (0.5 − 0.866i)9-s + (0.0950 − 0.995i)12-s + (0.909 − 0.415i)13-s + (−0.327 − 0.945i)16-s + (−0.690 + 0.723i)17-s + (0.998 + 0.0475i)18-s + (0.723 − 0.690i)19-s + (0.945 − 0.327i)23-s + (0.928 − 0.371i)24-s + (0.786 + 0.618i)26-s + i·27-s + ⋯ |
L(s) = 1 | + (0.458 + 0.888i)2-s + (−0.866 + 0.5i)3-s + (−0.580 + 0.814i)4-s + (−0.841 − 0.540i)6-s + (−0.989 − 0.142i)8-s + (0.5 − 0.866i)9-s + (0.0950 − 0.995i)12-s + (0.909 − 0.415i)13-s + (−0.327 − 0.945i)16-s + (−0.690 + 0.723i)17-s + (0.998 + 0.0475i)18-s + (0.723 − 0.690i)19-s + (0.945 − 0.327i)23-s + (0.928 − 0.371i)24-s + (0.786 + 0.618i)26-s + i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8738059814 - 0.1287929416i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8738059814 - 0.1287929416i\) |
\(L(1)\) |
\(\approx\) |
\(0.7851419591 + 0.4386656353i\) |
\(L(1)\) |
\(\approx\) |
\(0.7851419591 + 0.4386656353i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.458 + 0.888i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (0.909 - 0.415i)T \) |
| 17 | \( 1 + (-0.690 + 0.723i)T \) |
| 19 | \( 1 + (0.723 - 0.690i)T \) |
| 23 | \( 1 + (0.945 - 0.327i)T \) |
| 29 | \( 1 + (0.959 + 0.281i)T \) |
| 31 | \( 1 + (0.995 - 0.0950i)T \) |
| 37 | \( 1 + (0.814 - 0.580i)T \) |
| 41 | \( 1 + (-0.841 - 0.540i)T \) |
| 43 | \( 1 + (-0.989 - 0.142i)T \) |
| 47 | \( 1 + (-0.998 + 0.0475i)T \) |
| 53 | \( 1 + (-0.945 - 0.327i)T \) |
| 59 | \( 1 + (-0.888 - 0.458i)T \) |
| 61 | \( 1 + (-0.0475 - 0.998i)T \) |
| 67 | \( 1 + (-0.998 - 0.0475i)T \) |
| 71 | \( 1 + (-0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.189 - 0.981i)T \) |
| 79 | \( 1 + (-0.928 - 0.371i)T \) |
| 83 | \( 1 + (-0.755 + 0.654i)T \) |
| 89 | \( 1 + (0.235 - 0.971i)T \) |
| 97 | \( 1 + (-0.989 - 0.142i)T \) |
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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
−18.4236964607456562341095134809, −17.99291274631385825673942820305, −17.23266093516088767759366399391, −16.369155493284620491531950430119, −15.73743292380013491980367527667, −14.9623082730071699724529343061, −13.90943259476146434742660723399, −13.53483444788890252951133489448, −12.97905653568573007393879328489, −12.09106530393714352818012858969, −11.53434708301683225346174287861, −11.22880327085529031274138907991, −10.27260929133904416130347531633, −9.77985535183157522566067545542, −8.81057923960284627191780167995, −8.07293621921382898329051676942, −6.98022780685528173941717012040, −6.349458082414648141661191484595, −5.73280806812177187357407658374, −4.789041886317565391739892178, −4.46088662646526371560002993822, −3.27758534783922990194074267415, −2.625976843470819936638954869863, −1.419173318297913040147783354092, −1.151844918415189519204276382930,
0.26709006829140195808711015279, 1.38093888437259071213474678989, 2.955125871328967370006751219895, 3.508845885020344453130437324974, 4.53471075495475286206998520680, 4.86436750994113709180964811872, 5.741556965577082527036530068891, 6.43765094783461018096394214687, 6.81215832300096108544456027733, 7.856766021098075854645935897197, 8.626037566352296803730575861852, 9.23078211919429159208970764231, 10.1203837964587939311028881829, 10.90196877578331120781385391082, 11.537972762821631304625408549151, 12.27900606764116316519957258003, 13.06161475527366235032286659226, 13.47999814032819344108039305029, 14.46679072974636449354704188301, 15.18446247249364867730755318567, 15.7070212806497885704906539162, 16.1339749937652158074601981885, 17.00307699759460407800508951702, 17.45210964758278326784507870378, 18.07688326440098950901598897022