L(s) = 1 | + (0.755 − 0.654i)3-s + (−0.959 − 0.281i)5-s + (−0.281 + 0.959i)7-s + (0.142 − 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.755 − 0.654i)13-s + (−0.909 + 0.415i)15-s + (−0.415 + 0.909i)17-s + (0.989 + 0.142i)19-s + (0.415 + 0.909i)21-s + (0.989 + 0.142i)23-s + (0.841 + 0.540i)25-s + (−0.540 − 0.841i)27-s + (0.281 − 0.959i)29-s + (0.989 − 0.142i)31-s + ⋯ |
L(s) = 1 | + (0.755 − 0.654i)3-s + (−0.959 − 0.281i)5-s + (−0.281 + 0.959i)7-s + (0.142 − 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.755 − 0.654i)13-s + (−0.909 + 0.415i)15-s + (−0.415 + 0.909i)17-s + (0.989 + 0.142i)19-s + (0.415 + 0.909i)21-s + (0.989 + 0.142i)23-s + (0.841 + 0.540i)25-s + (−0.540 − 0.841i)27-s + (0.281 − 0.959i)29-s + (0.989 − 0.142i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6608648005 - 1.240567984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6608648005 - 1.240567984i\) |
\(L(1)\) |
\(\approx\) |
\(1.011451079 - 0.2937554307i\) |
\(L(1)\) |
\(\approx\) |
\(1.011451079 - 0.2937554307i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.755 - 0.654i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (-0.281 + 0.959i)T \) |
| 11 | \( 1 + (-0.959 + 0.281i)T \) |
| 13 | \( 1 + (0.755 - 0.654i)T \) |
| 17 | \( 1 + (-0.415 + 0.909i)T \) |
| 19 | \( 1 + (0.989 + 0.142i)T \) |
| 23 | \( 1 + (0.989 + 0.142i)T \) |
| 29 | \( 1 + (0.281 - 0.959i)T \) |
| 31 | \( 1 + (0.989 - 0.142i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.755 - 0.654i)T \) |
| 43 | \( 1 + (-0.281 - 0.959i)T \) |
| 47 | \( 1 + (-0.654 + 0.755i)T \) |
| 53 | \( 1 + (-0.654 - 0.755i)T \) |
| 59 | \( 1 + (0.755 + 0.654i)T \) |
| 61 | \( 1 + (-0.540 - 0.841i)T \) |
| 67 | \( 1 + (-0.654 - 0.755i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (-0.142 - 0.989i)T \) |
| 79 | \( 1 + (-0.142 - 0.989i)T \) |
| 83 | \( 1 + (0.909 + 0.415i)T \) |
| 97 | \( 1 + (-0.959 - 0.281i)T \) |
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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
−22.83855267116651129156962014387, −21.777273078132041446912952776301, −20.83924683042846217400370392401, −20.30066245569480815654793884399, −19.55095004343977151545839159907, −18.80441593496357555469040190383, −17.96952222351336593531251422302, −16.41411034975892535650415866815, −16.17334665853313780271033614295, −15.42338157439156036668321722827, −14.420808141522966879213759800016, −13.67869516665750308603523131166, −13.02772169526623798514403346260, −11.564603934511033478482384129299, −10.9256519285092227790823014373, −10.171473807948807246767725201907, −9.14780995026094488004174012790, −8.287413300517194723322375309927, −7.445780010941848258136291357, −6.75448789267478806877410798475, −5.06914320981884403817855211445, −4.35904608544759905536921995252, −3.34063450228884908438571719271, −2.82631901626305643060049531669, −1.05624190836148203699527150793,
0.330980331222144371660868021121, 1.58797226698260880051300728971, 2.85617222231410308406122610157, 3.42352308139428026746268790017, 4.74041547012501719613519986210, 5.8465780086829772818135616158, 6.85883528581120395914580401343, 7.97932392311880999564475849998, 8.30217396598311020705455464243, 9.214398543503214785583664694877, 10.32262524454894865409166083265, 11.53415884914728304567500855933, 12.25096918619273251626978553357, 13.020800317390776498279607956974, 13.596624582380678318190169183247, 15.14186582131380319544181532851, 15.271984177291134891204762770192, 16.05383011189080915885198863, 17.44416890742716161054859833124, 18.277075977310507396761133913035, 18.99840255408593556636252318639, 19.48143250254145436216393761409, 20.63786009899161871298426986884, 20.85274948620463216942276786701, 22.24533445868291360993291182518