L(s) = 1 | + (−0.540 − 0.841i)3-s + (−0.281 − 0.959i)7-s + (−0.415 + 0.909i)9-s + (0.654 + 0.755i)11-s + (0.281 − 0.959i)13-s + (0.989 + 0.142i)17-s + (−0.142 − 0.989i)19-s + (−0.654 + 0.755i)21-s + (0.989 − 0.142i)27-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (0.281 − 0.959i)33-s + (−0.909 − 0.415i)37-s + (−0.959 + 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯ |
L(s) = 1 | + (−0.540 − 0.841i)3-s + (−0.281 − 0.959i)7-s + (−0.415 + 0.909i)9-s + (0.654 + 0.755i)11-s + (0.281 − 0.959i)13-s + (0.989 + 0.142i)17-s + (−0.142 − 0.989i)19-s + (−0.654 + 0.755i)21-s + (0.989 − 0.142i)27-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (0.281 − 0.959i)33-s + (−0.909 − 0.415i)37-s + (−0.959 + 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4481289438 - 0.8279899438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4481289438 - 0.8279899438i\) |
\(L(1)\) |
\(\approx\) |
\(0.7713804654 - 0.4012398567i\) |
\(L(1)\) |
\(\approx\) |
\(0.7713804654 - 0.4012398567i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.540 - 0.841i)T \) |
| 7 | \( 1 + (-0.281 - 0.959i)T \) |
| 11 | \( 1 + (0.654 + 0.755i)T \) |
| 13 | \( 1 + (0.281 - 0.959i)T \) |
| 17 | \( 1 + (0.989 + 0.142i)T \) |
| 19 | \( 1 + (-0.142 - 0.989i)T \) |
| 29 | \( 1 + (0.142 - 0.989i)T \) |
| 31 | \( 1 + (-0.841 - 0.540i)T \) |
| 37 | \( 1 + (-0.909 - 0.415i)T \) |
| 41 | \( 1 + (0.415 + 0.909i)T \) |
| 43 | \( 1 + (-0.540 - 0.841i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.281 - 0.959i)T \) |
| 59 | \( 1 + (-0.959 - 0.281i)T \) |
| 61 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (-0.755 - 0.654i)T \) |
| 71 | \( 1 + (0.654 - 0.755i)T \) |
| 73 | \( 1 + (0.989 - 0.142i)T \) |
| 79 | \( 1 + (-0.959 - 0.281i)T \) |
| 83 | \( 1 + (-0.909 - 0.415i)T \) |
| 89 | \( 1 + (-0.841 + 0.540i)T \) |
| 97 | \( 1 + (0.909 - 0.415i)T \) |
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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
−24.11425516964484703350711778537, −23.2204684779293498576854151966, −22.41129017249103147098462539707, −21.55137597159770109513576820064, −21.21495392222705779307980265677, −20.025940918784789181696546719117, −18.89488055084901290076031256896, −18.35927332426209293983254379696, −17.0440352939485897990708279841, −16.413061534841614951357822744886, −15.79263474308000378270115121897, −14.64697795484555600926036765044, −14.0903102206139359877497054611, −12.49573388669612687360781953630, −11.89894642327953092159315375458, −11.05429039596564414581199050152, −10.01826229571480191365264121241, −9.122614155628260386339797330535, −8.48821339397436989964461636642, −6.830903628512045650947044640634, −5.92130131834341942757542703326, −5.229000915596590666003853189304, −3.90586357426315231042160040636, −3.15141304621305400744424238025, −1.48150780169409042387961770204,
0.60354667116656753767360994398, 1.74374249219523286843659659781, 3.1684914446843456810666571062, 4.40177010674660511099905075321, 5.5499452675177219315176584088, 6.55556895556000554869900622565, 7.34753141176729938430531622517, 8.10501773725053129793834706038, 9.532695214137653634387628155412, 10.482512580613864494587891211556, 11.3217056674659898128409939664, 12.35810815926708352609417750342, 13.05190817279728499956428878507, 13.85577329562585663771047556613, 14.856151707762813510844806743256, 16.01930613952677383800860507103, 17.065617094426837720873346791163, 17.448721027910741950295185904766, 18.40359909069960224702254557807, 19.46182785630357501743835208578, 19.97947352418840196610025922309, 20.978043056240426892765201836265, 22.33259589901041222256977557279, 22.83445161290166261952796390185, 23.55539366795401490731129538753