L(s) = 1 | + (0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (0.309 − 0.951i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (0.309 − 0.951i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2974799917 - 0.9177013403i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2974799917 - 0.9177013403i\) |
\(L(1)\) |
\(\approx\) |
\(0.6986263381 - 0.8009032803i\) |
\(L(1)\) |
\(\approx\) |
\(0.6986263381 - 0.8009032803i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 \) |
good | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (-0.809 + 0.587i)T \) |
| 47 | \( 1 + (0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + T \) |
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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
−32.10018650011734475736139928174, −31.42831548319946581081502180689, −30.62374414424634006299856221809, −28.30655318752459069895599718210, −27.54614711448784879065864067882, −26.83775036886913405610573497883, −25.177032448232243538822786609336, −24.96249828278937319827960967812, −23.15868206456786392862310139650, −22.54277729925778067201225488844, −21.15215637192293311191319668666, −20.25651781441391918541220777499, −18.60289000493752789175951252196, −17.1956044738936899529411353078, −16.02677991522209118319243258432, −15.3139953220971887837190211858, −14.49648311516042901080852136404, −12.79685706117192022938737428197, −11.65117241690968121374018134417, −9.64361169879213407340333752563, −8.61472145040845506246531181564, −7.61541582264587777157557126419, −5.52669789147003023966384543223, −4.64827047902065854909872442234, −3.27526540442796136980518357982,
1.16632299339860574045361892331, 3.00918260645587926930266165330, 4.154601741612806550822808282955, 6.313883352341728818566169892729, 7.70442989448004983546879258074, 9.02810248030520918818407323062, 10.98233413886806417607432941097, 11.49480199586583148993119832595, 12.9828863127665789505144155198, 13.96849738145078313089766039401, 14.84007173038876433723245487607, 16.93360603853131472167600354235, 18.35358203491721237597861963068, 19.234397234252233101571805702374, 19.87336433252962294469143743976, 21.1763625507074818310924855043, 22.56046102111023124572537638481, 23.72018266739959417747479084289, 24.0517658405000330562886212288, 26.17407785718693884588130584813, 26.8904551860661989669528120037, 28.25332594881897827597508935663, 29.51594104369282481386478896374, 30.23153083076018423443950361718, 30.925519612913097969183744693998