L(s) = 1 | + (−0.587 + 0.809i)2-s + (0.165 + 0.227i)3-s + (−0.309 − 0.951i)4-s + (−2.22 + 0.192i)5-s − 0.281·6-s + (3.62 − 1.17i)7-s + (0.951 + 0.309i)8-s + (0.902 − 2.77i)9-s + (1.15 − 1.91i)10-s + (−0.493 − 1.51i)11-s + (0.165 − 0.227i)12-s + (1.88 + 2.59i)13-s + (−1.17 + 3.62i)14-s + (−0.412 − 0.475i)15-s + (−0.809 + 0.587i)16-s + (5.38 + 1.74i)17-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.572i)2-s + (0.0955 + 0.131i)3-s + (−0.154 − 0.475i)4-s + (−0.996 + 0.0862i)5-s − 0.114·6-s + (1.36 − 0.444i)7-s + (0.336 + 0.109i)8-s + (0.300 − 0.925i)9-s + (0.364 − 0.605i)10-s + (−0.148 − 0.458i)11-s + (0.0477 − 0.0657i)12-s + (0.523 + 0.720i)13-s + (−0.314 + 0.968i)14-s + (−0.106 − 0.122i)15-s + (−0.202 + 0.146i)16-s + (1.30 + 0.424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08652 + 0.155546i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08652 + 0.155546i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 + (2.22 - 0.192i)T \) |
| 31 | \( 1 + (-4.81 + 2.80i)T \) |
good | 3 | \( 1 + (-0.165 - 0.227i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (-3.62 + 1.17i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (0.493 + 1.51i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.88 - 2.59i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.38 - 1.74i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.03 - 1.47i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.41 + 1.43i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.33 - 0.969i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + 1.74iT - 37T^{2} \) |
| 41 | \( 1 + (7.29 + 5.30i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.0681 + 0.0937i)T + (-13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-5.45 - 7.51i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (5.57 + 1.81i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.25 + 5.27i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 0.561iT - 67T^{2} \) |
| 71 | \( 1 + (-0.785 + 2.41i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.99 - 1.29i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.88 - 8.88i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (9.27 - 12.7i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.58 + 14.1i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (5.42 - 1.76i)T + (78.4 - 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.64966443727665906550643053255, −10.79387052766633871403501548731, −9.816395385440402065847593683084, −8.540543857095230839346551797588, −8.018309557645384301973078857625, −7.12587391536141163830945080678, −5.92392738404114507514548822205, −4.54620663659694989995717137218, −3.64267579670079372877128110989, −1.18639694488516160109192114010,
1.44450912466096286892429676658, 2.99665357923575571231940956096, 4.47590026936746332122559487707, 5.33589793400454754036968184323, 7.35285593344252112783022673665, 8.003806761213877669332407812917, 8.504050996723507415343521714683, 9.992541271347141055324350883233, 10.77346998681344449860723074404, 11.75608904832434018633770969694