L(s) = 1 | + (0.969 + 0.243i)2-s + (0.881 + 0.473i)4-s + (−0.650 − 0.759i)5-s + (0.952 + 0.303i)7-s + (0.739 + 0.673i)8-s + (−0.445 − 0.895i)10-s + (0.696 − 0.717i)11-s + (0.850 + 0.526i)14-s + (0.552 + 0.833i)16-s + (0.779 + 0.626i)17-s + (−0.992 + 0.122i)19-s + (−0.213 − 0.976i)20-s + (0.850 − 0.526i)22-s + (0.273 − 0.961i)23-s + (−0.153 + 0.988i)25-s + ⋯ |
L(s) = 1 | + (0.969 + 0.243i)2-s + (0.881 + 0.473i)4-s + (−0.650 − 0.759i)5-s + (0.952 + 0.303i)7-s + (0.739 + 0.673i)8-s + (−0.445 − 0.895i)10-s + (0.696 − 0.717i)11-s + (0.850 + 0.526i)14-s + (0.552 + 0.833i)16-s + (0.779 + 0.626i)17-s + (−0.992 + 0.122i)19-s + (−0.213 − 0.976i)20-s + (0.850 − 0.526i)22-s + (0.273 − 0.961i)23-s + (−0.153 + 0.988i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.888954121 + 0.2416798029i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.888954121 + 0.2416798029i\) |
\(L(1)\) |
\(\approx\) |
\(2.097976576 + 0.1454280670i\) |
\(L(1)\) |
\(\approx\) |
\(2.097976576 + 0.1454280670i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 + (0.969 + 0.243i)T \) |
| 5 | \( 1 + (-0.650 - 0.759i)T \) |
| 7 | \( 1 + (0.952 + 0.303i)T \) |
| 11 | \( 1 + (0.696 - 0.717i)T \) |
| 17 | \( 1 + (0.779 + 0.626i)T \) |
| 19 | \( 1 + (-0.992 + 0.122i)T \) |
| 23 | \( 1 + (0.273 - 0.961i)T \) |
| 29 | \( 1 + (-0.332 + 0.943i)T \) |
| 31 | \( 1 + (0.445 - 0.895i)T \) |
| 37 | \( 1 + (0.0922 - 0.995i)T \) |
| 41 | \( 1 + (0.650 - 0.759i)T \) |
| 43 | \( 1 + (0.908 + 0.417i)T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.389 + 0.920i)T \) |
| 59 | \( 1 + (-0.952 + 0.303i)T \) |
| 61 | \( 1 + (0.932 - 0.361i)T \) |
| 67 | \( 1 + (0.213 - 0.976i)T \) |
| 71 | \( 1 + (-0.332 - 0.943i)T \) |
| 73 | \( 1 + (-0.982 - 0.183i)T \) |
| 79 | \( 1 + (-0.982 + 0.183i)T \) |
| 83 | \( 1 + (0.213 + 0.976i)T \) |
| 89 | \( 1 + (0.850 + 0.526i)T \) |
| 97 | \( 1 + (0.779 - 0.626i)T \) |
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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
−18.82117954872136456588010579967, −17.62198316637109188705822931114, −17.21238394606902447158569139975, −16.175598087304338629126238481366, −15.51111058823271202579662880841, −14.76845209295274165110793566610, −14.55395561649815501722745414070, −13.812164302499800131298625985218, −13.0383042112699722394503178290, −12.08836338734261921318547161688, −11.65496177246938448429885556344, −11.18134408857532296924834498891, −10.330457211282166557232941537102, −9.80442301594772742822769130147, −8.593460865628413549513874658359, −7.6451761517754785020339185553, −7.23015937087358864601807746257, −6.5191835295553123514240525063, −5.65289979997843739281422109472, −4.722960171881277557565511948387, −4.237049903710566860023955263657, −3.51722041557875908906331900521, −2.67301851424323527776556717628, −1.85412173803124277845538438009, −0.98221628832289171332959803936,
0.94958125010486798168121395744, 1.772943734339887471452095408516, 2.7189704654998613354592382587, 3.75779312209136552933165208086, 4.245241087148695619447132285702, 4.892542978208824395037590854771, 5.781741111308630529554478825, 6.21556910759340219918620379611, 7.403662327333488358621313997, 7.90350234654231973648425954750, 8.62012415161221895160010591341, 9.15477704953384412930152526079, 10.72123427266805583322073295981, 10.98167341440469844219248086051, 11.832680703463990467062018302531, 12.464919751283022561967849499185, 12.791738929874027011498145874818, 13.8698581076230968861584402729, 14.504911837766927977974365075294, 14.89654833655908353449253168260, 15.71720009650308207400568179366, 16.38704409072606995037993923210, 17.00237729448256446398837551076, 17.41296113130678538335927552502, 18.69320393469317070673947118285