| L(s) = 1 | − 0.561·2-s − 7.68·4-s + 3.31·5-s + 7·7-s + 8.80·8-s − 1.86·10-s − 11·11-s − 41.9·13-s − 3.93·14-s + 56.5·16-s + 68.8·17-s − 114.·19-s − 25.4·20-s + 6.17·22-s + 124.·23-s − 114.·25-s + 23.5·26-s − 53.7·28-s + 147.·29-s − 55.9·31-s − 102.·32-s − 38.6·34-s + 23.2·35-s + 162.·37-s + 64.2·38-s + 29.2·40-s − 258.·41-s + ⋯ |
| L(s) = 1 | − 0.198·2-s − 0.960·4-s + 0.296·5-s + 0.377·7-s + 0.389·8-s − 0.0588·10-s − 0.301·11-s − 0.894·13-s − 0.0750·14-s + 0.883·16-s + 0.982·17-s − 1.38·19-s − 0.284·20-s + 0.0598·22-s + 1.13·23-s − 0.912·25-s + 0.177·26-s − 0.363·28-s + 0.945·29-s − 0.324·31-s − 0.564·32-s − 0.195·34-s + 0.112·35-s + 0.724·37-s + 0.274·38-s + 0.115·40-s − 0.985·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.305598327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.305598327\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.561T + 8T^{2} \) |
| 5 | \( 1 - 3.31T + 125T^{2} \) |
| 13 | \( 1 + 41.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 55.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 258.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 106.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 10.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 189.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 580.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 79.6T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 874.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 844.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 925.T + 9.12e5T^{2} \) |
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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
−10.00193232648248253138585710998, −9.262291741723733578835309238229, −8.329262106372243706091523440451, −7.71149394828181567647543638257, −6.52097645129530690922416049163, −5.30800327038205900887971515050, −4.73033359160660884453408105521, −3.54747174217649852290930377969, −2.13387747067743997429406781782, −0.68492672600460339496107357010,
0.68492672600460339496107357010, 2.13387747067743997429406781782, 3.54747174217649852290930377969, 4.73033359160660884453408105521, 5.30800327038205900887971515050, 6.52097645129530690922416049163, 7.71149394828181567647543638257, 8.329262106372243706091523440451, 9.262291741723733578835309238229, 10.00193232648248253138585710998
