| L(s) = 1 | + 4.35·2-s + 11.0·4-s + 8.71·5-s − 7·7-s + 13.0·8-s + 38.0·10-s − 43.5·11-s + 82·13-s − 30.5·14-s − 30.9·16-s − 78.4·17-s − 20·19-s + 95.8·20-s − 190.·22-s − 130.·23-s − 48.9·25-s + 357.·26-s − 77.0·28-s + 244.·29-s + 156·31-s − 239.·32-s − 342.·34-s − 61.0·35-s + 186·37-s − 87.1·38-s + 114.·40-s − 165.·41-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.37·4-s + 0.779·5-s − 0.377·7-s + 0.577·8-s + 1.20·10-s − 1.19·11-s + 1.74·13-s − 0.582·14-s − 0.484·16-s − 1.11·17-s − 0.241·19-s + 1.07·20-s − 1.84·22-s − 1.18·23-s − 0.391·25-s + 2.69·26-s − 0.519·28-s + 1.56·29-s + 0.903·31-s − 1.32·32-s − 1.72·34-s − 0.294·35-s + 0.826·37-s − 0.372·38-s + 0.450·40-s − 0.630·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.086671303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.086671303\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 4.35T + 8T^{2} \) |
| 5 | \( 1 - 8.71T + 125T^{2} \) |
| 11 | \( 1 + 43.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 156T + 2.97e4T^{2} \) |
| 37 | \( 1 - 186T + 5.06e4T^{2} \) |
| 41 | \( 1 + 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164T + 7.95e4T^{2} \) |
| 47 | \( 1 - 470.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 156.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 156.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 790T + 2.26e5T^{2} \) |
| 67 | \( 1 + 44T + 3.00e5T^{2} \) |
| 71 | \( 1 - 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 126T + 3.89e5T^{2} \) |
| 79 | \( 1 + 712T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 798T + 9.12e5T^{2} \) |
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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
−13.96785455537949929393881666455, −13.50640275149007135629872319687, −12.69989022722962218748552327117, −11.33541398420770267525710753026, −10.18051851977067388998539362168, −8.498283235177754701987777796728, −6.48476799588701545015873416061, −5.70392303032643907704309463931, −4.18295387428861661860873926800, −2.54156397125768629133131145321,
2.54156397125768629133131145321, 4.18295387428861661860873926800, 5.70392303032643907704309463931, 6.48476799588701545015873416061, 8.498283235177754701987777796728, 10.18051851977067388998539362168, 11.33541398420770267525710753026, 12.69989022722962218748552327117, 13.50640275149007135629872319687, 13.96785455537949929393881666455
