L(s) = 1 | + (−0.124 − 0.0157i)3-s + (−0.187 − 0.982i)4-s + (−0.809 − 0.587i)5-s + (−0.953 − 0.244i)9-s + (−0.637 + 0.770i)11-s + (0.00788 + 0.125i)12-s + (0.0915 + 0.0859i)15-s + (−0.929 + 0.368i)16-s + (−0.425 + 0.904i)20-s + (−1.06 − 1.67i)23-s + (0.309 + 0.951i)25-s + (0.231 + 0.0917i)27-s + (−0.200 + 1.05i)31-s + (0.0915 − 0.0859i)33-s + (−0.0618 + 0.982i)36-s + (−0.574 + 0.227i)37-s + ⋯ |
L(s) = 1 | + (−0.124 − 0.0157i)3-s + (−0.187 − 0.982i)4-s + (−0.809 − 0.587i)5-s + (−0.953 − 0.244i)9-s + (−0.637 + 0.770i)11-s + (0.00788 + 0.125i)12-s + (0.0915 + 0.0859i)15-s + (−0.929 + 0.368i)16-s + (−0.425 + 0.904i)20-s + (−1.06 − 1.67i)23-s + (0.309 + 0.951i)25-s + (0.231 + 0.0917i)27-s + (−0.200 + 1.05i)31-s + (0.0915 − 0.0859i)33-s + (−0.0618 + 0.982i)36-s + (−0.574 + 0.227i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2427185878\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2427185878\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.637 - 0.770i)T \) |
good | 2 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 3 | \( 1 + (0.124 + 0.0157i)T + (0.968 + 0.248i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 17 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 19 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 23 | \( 1 + (1.06 + 1.67i)T + (-0.425 + 0.904i)T^{2} \) |
| 29 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 31 | \( 1 + (0.200 - 1.05i)T + (-0.929 - 0.368i)T^{2} \) |
| 37 | \( 1 + (0.574 - 0.227i)T + (0.728 - 0.684i)T^{2} \) |
| 41 | \( 1 + (0.425 + 0.904i)T^{2} \) |
| 43 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 47 | \( 1 + (1.41 - 0.779i)T + (0.535 - 0.844i)T^{2} \) |
| 53 | \( 1 + (0.929 + 0.872i)T + (0.0627 + 0.998i)T^{2} \) |
| 59 | \( 1 + (0.116 + 1.85i)T + (-0.992 + 0.125i)T^{2} \) |
| 61 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 67 | \( 1 + (0.620 + 1.31i)T + (-0.637 + 0.770i)T^{2} \) |
| 71 | \( 1 + (-0.110 + 0.0604i)T + (0.535 - 0.844i)T^{2} \) |
| 73 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 79 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 83 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 89 | \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \) |
| 97 | \( 1 + (-0.791 + 1.68i)T + (-0.637 - 0.770i)T^{2} \) |
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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
−9.333225702245841614023833238040, −8.541984208244744107528480951074, −7.928310452495021302208600630608, −6.76814399227617481877911551569, −6.00592931616038944707113455329, −4.96991587942808371548394650466, −4.56377655606696476937074181484, −3.20434475157921129653507165504, −1.84112987206505368754469864201, −0.19351589102559275432015516175,
2.46256272595576223407420774081, 3.33061068382849150894716030597, 3.97968233166823522946602992928, 5.22569050021758744692946993948, 6.08997963278246694194342808696, 7.21226467002356862190807426968, 7.902961388410951019796766541616, 8.332406479863926065516855417918, 9.235508093108605283931440327203, 10.34038538800422737398710831972