L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s + i·12-s + i·13-s + 14-s + 16-s + i·17-s + i·18-s + 21-s + i·23-s + 24-s + ⋯ |
L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s + i·12-s + i·13-s + 14-s + 16-s + i·17-s + i·18-s + 21-s + i·23-s + 24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.08618615524 - 0.3033879376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.08618615524 - 0.3033879376i\) |
\(L(1)\) |
\(\approx\) |
\(0.6613525516 - 0.4087383554i\) |
\(L(1)\) |
\(\approx\) |
\(0.6613525516 - 0.4087383554i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 \) |
| 73 | \( 1 \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + T \) |
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
−22.06737056595351825533794427088, −21.02871815548059020420452911873, −20.33848676068962295944799147355, −19.613833017490878490580726542651, −18.38354076838746714388005799329, −17.66754080553940393621668042132, −16.90010516232562469085011823011, −16.27047981820202343507485790627, −15.717570274442580404638471570978, −14.66781935167394634798916784370, −14.32604746301232828719538313113, −13.34033906609519775159858542816, −12.55218852211099252211054953167, −11.12286821296966415828372870200, −10.50364388431273847196158224412, −9.61856686813930107902944530222, −8.999860383502215931236175702211, −7.89953373447294290319579148347, −7.32699992978469771905430394320, −6.16258262070617627231157381605, −5.35823916746558109031053442824, −4.52685577250632722474432662514, −3.81888768068722673082611854080, −2.84535770606721246302611092890, −0.81987859932925604278075656713,
0.08370402013394201195611241920, 1.60855084693943211680781341038, 1.95949988121538654046342061260, 3.05944673299176247725057133799, 4.00532956450267960053830033603, 5.33095184669005947341470578947, 5.93019460132012839964409315581, 7.08994386754290259276632866646, 8.13666498576018543069313635498, 8.86978352701435215817134723218, 9.499539338525112713939686397935, 10.73306505060683261794589835531, 11.63008886864186032925074226156, 11.98267924050323664249643677934, 12.94143230205821033606709371140, 13.407854061382781485787824887896, 14.459620537016858369441286420156, 15.02152864894046874115843644613, 16.44135455258174925121068411722, 17.341541466504074546161609185627, 18.00400609801402791389796590598, 18.88289874252776414767936484258, 19.09855926942654611959922282964, 19.97081370200061708782074842028, 20.84914881432352387961734757177