L(s) = 1 | + (0.222 + 0.974i)3-s + (−0.900 + 0.433i)9-s + (0.433 − 0.900i)11-s + (−0.900 − 0.433i)13-s + (0.781 + 0.623i)17-s − i·19-s + (−0.781 + 0.623i)23-s + (−0.623 − 0.781i)27-s + (−0.781 − 0.623i)29-s + 31-s + (0.974 + 0.222i)33-s + (−0.623 + 0.781i)37-s + (0.222 − 0.974i)39-s + (−0.222 − 0.974i)41-s + (0.222 − 0.974i)43-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)3-s + (−0.900 + 0.433i)9-s + (0.433 − 0.900i)11-s + (−0.900 − 0.433i)13-s + (0.781 + 0.623i)17-s − i·19-s + (−0.781 + 0.623i)23-s + (−0.623 − 0.781i)27-s + (−0.781 − 0.623i)29-s + 31-s + (0.974 + 0.222i)33-s + (−0.623 + 0.781i)37-s + (0.222 − 0.974i)39-s + (−0.222 − 0.974i)41-s + (0.222 − 0.974i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6080897783 + 1.078980496i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6080897783 + 1.078980496i\) |
\(L(1)\) |
\(\approx\) |
\(0.9549032164 + 0.2545091124i\) |
\(L(1)\) |
\(\approx\) |
\(0.9549032164 + 0.2545091124i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.222 + 0.974i)T \) |
| 11 | \( 1 + (0.433 - 0.900i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| 17 | \( 1 + (0.781 + 0.623i)T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (-0.781 + 0.623i)T \) |
| 29 | \( 1 + (-0.781 - 0.623i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.623 + 0.781i)T \) |
| 41 | \( 1 + (-0.222 - 0.974i)T \) |
| 43 | \( 1 + (0.222 - 0.974i)T \) |
| 47 | \( 1 + (-0.433 + 0.900i)T \) |
| 53 | \( 1 + (0.623 + 0.781i)T \) |
| 59 | \( 1 + (-0.974 - 0.222i)T \) |
| 61 | \( 1 + (-0.781 - 0.623i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.433 - 0.900i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.900 - 0.433i)T \) |
| 97 | \( 1 + iT \) |
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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
−18.15989421262820697588229564640, −17.6208354422352312501938675903, −16.689200204787037797998925819356, −16.41806712725579839799837581759, −15.08431189690480496076554232040, −14.60796686626585819011991457339, −14.10448512137720715658388269586, −13.34279605713307672596711005615, −12.357244934866139891580338716804, −12.19870293090193089278117536720, −11.5423652907449193006729900983, −10.40818383828488536539090036118, −9.70324837938287355949401891863, −9.10552033111543727672517135198, −8.106517539467467030085541842252, −7.60687840298099200534398818125, −6.91666063033767431220194263349, −6.28266271419575351194203497452, −5.40727389536507270543534675228, −4.58091538591139763553730934143, −3.65152975426533828802982158539, −2.76240788665227455018159042396, −1.9600921123840943431243066015, −1.354563124029321202797417359654, −0.23181357098742737875832474365,
0.66963654628212465497951442886, 1.91984782114920167147225805281, 2.880554484175290435644372923218, 3.47522838368130963530452704162, 4.23620322720501271409733189375, 5.06655596151306367018514721166, 5.71292901595539377662323765344, 6.4309115338137843454863271194, 7.61544534440962584866721928008, 8.1050229496513812940332114483, 9.02357298263318395459423899952, 9.48532784326943292820907796588, 10.35914493314783313381313833768, 10.75962597196484142251455808461, 11.7736026021827674965516576155, 12.12631977382227743311682953974, 13.367190818305863568612056451634, 13.85470609759867783805155209708, 14.559842352994254210316893499030, 15.31054244697941486161890805865, 15.68485105892122651225430777052, 16.59838149552581539057607192024, 17.18150914861726461771029768071, 17.55682802453144039792116491559, 18.86029606651020217115934316899