L(s) = 1 | + (0.587 − 0.809i)2-s + (0.309 − 0.951i)3-s + (−0.309 − 0.951i)4-s + (0.587 + 0.809i)5-s + (−0.587 − 0.809i)6-s + (0.951 − 0.309i)7-s + (−0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + 10-s − 12-s + (0.309 − 0.951i)14-s + (0.951 − 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.951 + 0.309i)18-s + (0.951 + 0.309i)19-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)2-s + (0.309 − 0.951i)3-s + (−0.309 − 0.951i)4-s + (0.587 + 0.809i)5-s + (−0.587 − 0.809i)6-s + (0.951 − 0.309i)7-s + (−0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + 10-s − 12-s + (0.309 − 0.951i)14-s + (0.951 − 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.951 + 0.309i)18-s + (0.951 + 0.309i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9707735065 - 1.397141237i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9707735065 - 1.397141237i\) |
\(L(1)\) |
\(\approx\) |
\(1.211249958 - 1.000581543i\) |
\(L(1)\) |
\(\approx\) |
\(1.211249958 - 1.000581543i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.587 + 0.809i)T \) |
| 7 | \( 1 + (0.951 - 0.309i)T \) |
| 17 | \( 1 + (-0.809 + 0.587i)T \) |
| 19 | \( 1 + (0.951 + 0.309i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.587 + 0.809i)T \) |
| 37 | \( 1 + (-0.951 + 0.309i)T \) |
| 41 | \( 1 + (0.951 + 0.309i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.951 + 0.309i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.951 + 0.309i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (0.587 + 0.809i)T \) |
| 73 | \( 1 + (0.951 - 0.309i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 + iT \) |
| 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
−28.30535994283989517963746873723, −27.48162474831397996914745661313, −26.484354633940719946201454957093, −25.56034772603579049350994609220, −24.59679872412464469190635375910, −23.98899999297826503680984217832, −22.397608518034659337717718694208, −21.79451572081709430717732298388, −20.78903333610676334833025672972, −20.2251154253549143851084175186, −18.0874886491568709925610534642, −17.29527535590355061233750605636, −16.21304681506273876116001795406, −15.51877265973721679747465705608, −14.29523032567562811613737852294, −13.711544620212885428007398085978, −12.3070318196124142876885585143, −11.1086020342673518268081369309, −9.404786837964049563888248807877, −8.73656626957322425289591602990, −7.59467284679671539670238384346, −5.73939400227801243467517037452, −5.02081012811860514606590643250, −4.05687238167291273771312599671, −2.3741348056110019924672793090,
1.526433299841239666634110166307, 2.425052668006634599057543512098, 3.80607662232747198743761026819, 5.48964594953182135326530027883, 6.55208641733410707258323049604, 7.8559904260276820480673465979, 9.32044241226579780145065545245, 10.64794222417523569145493299472, 11.497879020873651665684601758633, 12.616639969396615596789595383694, 13.885569274262829200710811617153, 14.14171146076412207790501741421, 15.29229156478985210444341016112, 17.526130860781960619018343191245, 18.08038898639763807737741972330, 19.045400368848677166407012966808, 20.07944579754882302579464263185, 20.95655161645955884032585582110, 22.057069316517838310348501158324, 22.9761216730911839280807237621, 24.06313708158533464909247695119, 24.67750424620730997032057605811, 26.05237132756430142704168773295, 27.01971910655450572753097022404, 28.40547952153029459457411580318