L(s) = 1 | + (0.669 + 0.743i)3-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (0.809 + 0.587i)13-s + (−0.978 − 0.207i)17-s + (−0.669 + 0.743i)19-s + (0.913 + 0.406i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.978 − 0.207i)31-s + (0.669 − 0.743i)33-s + (−0.104 + 0.994i)37-s + (0.104 + 0.994i)39-s + (0.809 + 0.587i)41-s − 43-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)3-s + (−0.104 + 0.994i)9-s + (−0.104 − 0.994i)11-s + (0.809 + 0.587i)13-s + (−0.978 − 0.207i)17-s + (−0.669 + 0.743i)19-s + (0.913 + 0.406i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.978 − 0.207i)31-s + (0.669 − 0.743i)33-s + (−0.104 + 0.994i)37-s + (0.104 + 0.994i)39-s + (0.809 + 0.587i)41-s − 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9448030819 + 1.357818614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9448030819 + 1.357818614i\) |
\(L(1)\) |
\(\approx\) |
\(1.145643489 + 0.4766794561i\) |
\(L(1)\) |
\(\approx\) |
\(1.145643489 + 0.4766794561i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.669 + 0.743i)T \) |
| 11 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (-0.978 - 0.207i)T \) |
| 19 | \( 1 + (-0.669 + 0.743i)T \) |
| 23 | \( 1 + (0.913 + 0.406i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + (-0.104 + 0.994i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.978 - 0.207i)T \) |
| 53 | \( 1 + (0.669 + 0.743i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (0.978 + 0.207i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.104 - 0.994i)T \) |
| 79 | \( 1 + (0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−20.46576427923372933333227984485, −19.82176423401452880916287773075, −19.13142225246588240170688283932, −18.26066872704973191405986752579, −17.71371750823176465310022480154, −17.04469256139005859847334714372, −15.71766503850340638549221421563, −15.20824614741558052095926259951, −14.52812801881542574210501424341, −13.5102898939432644489692985567, −12.953008503891764412340565961000, −12.44472081731189688729160069570, −11.28267015998934267559262016720, −10.6166828190202183554366436603, −9.43922447160037300311307649043, −8.83899090283019102020894691909, −8.06123436276907876448638368685, −7.1488174428755070227844092637, −6.602760340258083157963787286746, −5.581334305689212259380348105406, −4.4205941291703399661243262134, −3.59024924140591980635152244734, −2.46400057282091247570793140992, −1.88303289816829331068833751882, −0.56940432017611691848775972052,
1.375959993213142453049105140411, 2.452438598713238653875948012385, 3.41857510386352918302725726310, 4.04173359699520478846943268375, 5.02067669959865646790512927387, 5.918949498522784011703410127175, 6.88949127943633679524203815156, 7.9384951896444702817974816148, 8.83772560315380386818011678119, 9.06221730132862054722410416056, 10.26321604324328747237472382988, 10.974322848431402269332979740712, 11.49178554772765930589727797095, 12.883154252705860484207693644623, 13.504819598547736798613324872733, 14.18328912026730752178593083449, 15.00405558729790077902765107631, 15.67388349804084109501694057593, 16.51559543542700060924336993272, 16.88504403270814486301972665819, 18.2517861080631885061366873550, 18.82018834015388250911147136360, 19.59258999222255839346082629162, 20.36075193028240758061160812049, 21.04206608941831311104854372078