L(s) = 1 | + (0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)6-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)12-s + (−0.951 + 0.309i)13-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s − i·18-s + (−0.809 + 0.587i)19-s + (0.587 − 0.809i)22-s + (−0.951 − 0.309i)23-s + 24-s + ⋯ |
L(s) = 1 | + (0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)6-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)12-s + (−0.951 + 0.309i)13-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s − i·18-s + (−0.809 + 0.587i)19-s + (0.587 − 0.809i)22-s + (−0.951 − 0.309i)23-s + 24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.272815633 - 0.2148443325i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.272815633 - 0.2148443325i\) |
\(L(1)\) |
\(\approx\) |
\(1.988713593 - 0.09698595871i\) |
\(L(1)\) |
\(\approx\) |
\(1.988713593 - 0.09698595871i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 3 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
| 17 | \( 1 + (0.587 + 0.809i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.951 - 0.309i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.587 + 0.809i)T \) |
| 53 | \( 1 + (-0.587 + 0.809i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + (-0.587 - 0.809i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.951 + 0.309i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (-0.587 + 0.809i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.68544386706406230307250885771, −26.44922138413990911240733037838, −25.3167295432202361039037360329, −24.75496537973176608986600907137, −23.34759535781923596561649595052, −22.512113904771316555894184243823, −21.659383606762648676383322979317, −20.83142817608857146750227912644, −19.901469072490898084612948215573, −19.31265528674426796671207046570, −17.55580276372943876167354530415, −16.27879637862939865556818482015, −15.353890403892311257588232606140, −14.59857619480464016712800366095, −13.76188287917822886485942090532, −12.52666772658282097251445367294, −11.54859903052508344586366031859, −10.19415274465554174761911596716, −9.64666708050747968830468331530, −7.97014058918377996601633171495, −6.7186398726964573921882585160, −5.11856375570825935010342380983, −4.43476840186461125536923165558, −3.13551980171833803661258978433, −2.08623321940557213981087279511,
1.79653092832596328955307676052, 3.03497780846917617500535223117, 4.161111751455709667382412456697, 5.80122804329082864911015137405, 6.64485792091609597125126304833, 7.82436082531592242954669660570, 8.66341559813823002253200037470, 10.418177198638513088020260167550, 11.95413596942740743622890091109, 12.45094364052492590136641773235, 13.7200128333520777707172829943, 14.31096041510086897879668353233, 15.23016429073546519970986396351, 16.58271585050957449799681897075, 17.427049271536460712345948766336, 18.96450606581316805209329608537, 19.616961408814601021745630625378, 20.79159497079494242452689623422, 21.65251322438097432495238081431, 22.7072436327604306933419334752, 23.97413948359027790024653959916, 24.20499320312529385034668142921, 25.35147997867857280805942066917, 26.057772283114619794268302920596, 27.14924462018213692913427557764