L(s) = 1 | − 1.64·3-s − 5-s − 1.57·7-s − 0.297·9-s − 5.49·11-s − 2.44·13-s + 1.64·15-s − 4.78·17-s − 5.38·19-s + 2.58·21-s − 5.28·23-s + 25-s + 5.42·27-s + 9.94·29-s − 6.86·31-s + 9.02·33-s + 1.57·35-s + 37-s + 4.01·39-s − 7.92·41-s + 2.94·43-s + 0.297·45-s − 2.65·47-s − 4.52·49-s + 7.87·51-s + 3.79·53-s + 5.49·55-s + ⋯ |
L(s) = 1 | − 0.949·3-s − 0.447·5-s − 0.594·7-s − 0.0991·9-s − 1.65·11-s − 0.677·13-s + 0.424·15-s − 1.16·17-s − 1.23·19-s + 0.564·21-s − 1.10·23-s + 0.200·25-s + 1.04·27-s + 1.84·29-s − 1.23·31-s + 1.57·33-s + 0.265·35-s + 0.164·37-s + 0.643·39-s − 1.23·41-s + 0.448·43-s + 0.0443·45-s − 0.387·47-s − 0.646·49-s + 1.10·51-s + 0.521·53-s + 0.740·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1298386203\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1298386203\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 + 1.64T + 3T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 + 5.49T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 5.28T + 23T^{2} \) |
| 29 | \( 1 - 9.94T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 41 | \( 1 + 7.92T + 41T^{2} \) |
| 43 | \( 1 - 2.94T + 43T^{2} \) |
| 47 | \( 1 + 2.65T + 47T^{2} \) |
| 53 | \( 1 - 3.79T + 53T^{2} \) |
| 59 | \( 1 + 8.37T + 59T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 - 1.02T + 71T^{2} \) |
| 73 | \( 1 - 7.74T + 73T^{2} \) |
| 79 | \( 1 + 5.77T + 79T^{2} \) |
| 83 | \( 1 + 6.67T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.484466776528261714395616981148, −8.148650580432955340574303467765, −7.02209902942593685530128895815, −6.50141526033686215223738971486, −5.66144568214352724233835376471, −4.93064388110578164841454479875, −4.24365769883305460562549368284, −2.98334393633610603359478813246, −2.20480057418640594509433621264, −0.21168922304554128993411964627,
0.21168922304554128993411964627, 2.20480057418640594509433621264, 2.98334393633610603359478813246, 4.24365769883305460562549368284, 4.93064388110578164841454479875, 5.66144568214352724233835376471, 6.50141526033686215223738971486, 7.02209902942593685530128895815, 8.148650580432955340574303467765, 8.484466776528261714395616981148