| L(s) = 1 | + 8.50·2-s − 13.0·3-s + 40.3·4-s − 25·5-s − 110.·6-s − 217.·7-s + 70.7·8-s − 73.1·9-s − 212.·10-s − 525.·12-s − 747.·13-s − 1.84e3·14-s + 325.·15-s − 688.·16-s + 677.·17-s − 622.·18-s + 1.91e3·19-s − 1.00e3·20-s + 2.83e3·21-s − 3.51e3·23-s − 921.·24-s + 625·25-s − 6.35e3·26-s + 4.12e3·27-s − 8.76e3·28-s − 4.65e3·29-s + 2.77e3·30-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 0.835·3-s + 1.25·4-s − 0.447·5-s − 1.25·6-s − 1.67·7-s + 0.390·8-s − 0.301·9-s − 0.672·10-s − 1.05·12-s − 1.22·13-s − 2.52·14-s + 0.373·15-s − 0.672·16-s + 0.568·17-s − 0.452·18-s + 1.21·19-s − 0.563·20-s + 1.40·21-s − 1.38·23-s − 0.326·24-s + 0.200·25-s − 1.84·26-s + 1.08·27-s − 2.11·28-s − 1.02·29-s + 0.561·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.176010872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.176010872\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 8.50T + 32T^{2} \) |
| 3 | \( 1 + 13.0T + 243T^{2} \) |
| 7 | \( 1 + 217.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 747.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 677.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.91e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 371.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.63e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.24e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.71e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.35e4T + 8.58e9T^{2} \) |
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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
−9.956475750587842180707568165813, −9.313332719586046985763561065770, −7.65775198407954555284312305972, −6.81199438016759557304238588559, −5.89842577631873484568614518352, −5.45987913902599921645016655025, −4.29544872522570865616487132370, −3.36632281184492343595768238307, −2.59581071956584402686112531180, −0.40526408301731041585680161084,
0.40526408301731041585680161084, 2.59581071956584402686112531180, 3.36632281184492343595768238307, 4.29544872522570865616487132370, 5.45987913902599921645016655025, 5.89842577631873484568614518352, 6.81199438016759557304238588559, 7.65775198407954555284312305972, 9.313332719586046985763561065770, 9.956475750587842180707568165813
