L(s) = 1 | + (−0.173 − 0.984i)5-s + (−0.984 − 0.173i)7-s + (0.766 + 0.642i)11-s + (−0.5 + 0.866i)17-s − 19-s + (−0.173 − 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.984 + 0.173i)29-s + (−0.642 − 0.766i)31-s − i·35-s − 37-s + (−0.642 − 0.766i)41-s + (0.342 + 0.939i)43-s + (0.642 − 0.766i)47-s + (0.939 + 0.342i)49-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)5-s + (−0.984 − 0.173i)7-s + (0.766 + 0.642i)11-s + (−0.5 + 0.866i)17-s − 19-s + (−0.173 − 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.984 + 0.173i)29-s + (−0.642 − 0.766i)31-s − i·35-s − 37-s + (−0.642 − 0.766i)41-s + (0.342 + 0.939i)43-s + (0.642 − 0.766i)47-s + (0.939 + 0.342i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7280802731 + 0.3345383315i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7280802731 + 0.3345383315i\) |
\(L(1)\) |
\(\approx\) |
\(0.8126806947 - 0.09340800951i\) |
\(L(1)\) |
\(\approx\) |
\(0.8126806947 - 0.09340800951i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| 11 | \( 1 + (0.766 + 0.642i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (0.984 + 0.173i)T \) |
| 31 | \( 1 + (-0.642 - 0.766i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.642 - 0.766i)T \) |
| 43 | \( 1 + (0.342 + 0.939i)T \) |
| 47 | \( 1 + (0.642 - 0.766i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (0.766 - 0.642i)T \) |
| 61 | \( 1 + (-0.984 - 0.173i)T \) |
| 67 | \( 1 + (-0.766 + 0.642i)T \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 + (0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.939 + 0.342i)T \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 97 | \( 1 + (0.342 + 0.939i)T \) |
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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
−17.76953147034317416681031828552, −17.11210563113476021858108873758, −16.3594564923171779636962764603, −15.65398088695548870073758096487, −15.292867476951397065847107420423, −14.36069402122033654129484862516, −13.80471747984550214241529727358, −13.32076397718916434108373803677, −12.20463337178527603999079464302, −11.91803494534310459374947681329, −10.94061249269774419328469681206, −10.55462663998825398564527289816, −9.68730525502232523964952817905, −9.0721429711742962652304790007, −8.43576243320698267853411851446, −7.36266718913812950962578779104, −6.87317936268762495938327188000, −6.25702812434674594472526320343, −5.68842306239055282119135958196, −4.564682176253931804498613316043, −3.73902069995457906593178787465, −3.168449632044518278198978175323, −2.54225030677372014391729940826, −1.55658429170651011911464485858, −0.27449145414267122593551695622,
0.72136847762719584560937334647, 1.74865865591376472334054766015, 2.41532073917319562467492006779, 3.66810391697528729026302115867, 4.0749837113773457649142589138, 4.75958651713891977268556769748, 5.68211233607125124300205773652, 6.50688887909403700926613103926, 6.8676598149040062089790506942, 7.92820012148789939315059327611, 8.70941162216781931809874363950, 9.02706693946010395734964631662, 9.96667379006454539521357057362, 10.40124344664436005297973507747, 11.365203947916056277605191616622, 12.20593223846574658951110078415, 12.65658053769442827657728585614, 13.06342791303166834695841239963, 13.91771946116529686528500407104, 14.69882727721227233734152872727, 15.40795788939404709357364836183, 15.97389461567016576994234389084, 16.736480655502481662827446239810, 17.112969096790292256223448488891, 17.70247905684796500200125675415