| L(s) = 1 | − 1.79·2-s + 1.22·4-s − 1.34·5-s − 1.11·7-s + 1.39·8-s + 2.41·10-s + 2.46·11-s − 2.81·13-s + 1.99·14-s − 4.95·16-s − 5.50·17-s − 0.353·19-s − 1.64·20-s − 4.43·22-s − 5.31·23-s − 3.19·25-s + 5.05·26-s − 1.36·28-s − 4.45·31-s + 6.08·32-s + 9.88·34-s + 1.49·35-s − 9.62·37-s + 0.634·38-s − 1.88·40-s + 9.05·41-s − 9.04·43-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.610·4-s − 0.601·5-s − 0.421·7-s + 0.494·8-s + 0.763·10-s + 0.744·11-s − 0.781·13-s + 0.534·14-s − 1.23·16-s − 1.33·17-s − 0.0811·19-s − 0.367·20-s − 0.944·22-s − 1.10·23-s − 0.638·25-s + 0.991·26-s − 0.257·28-s − 0.799·31-s + 1.07·32-s + 1.69·34-s + 0.253·35-s − 1.58·37-s + 0.102·38-s − 0.297·40-s + 1.41·41-s − 1.38·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2177330719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2177330719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 5.50T + 17T^{2} \) |
| 19 | \( 1 + 0.353T + 19T^{2} \) |
| 23 | \( 1 + 5.31T + 23T^{2} \) |
| 31 | \( 1 + 4.45T + 31T^{2} \) |
| 37 | \( 1 + 9.62T + 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 + 9.04T + 43T^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 61 | \( 1 - 0.118T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 7.50T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 - 0.257T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 + 7.50T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−8.028106340670456030923206080284, −7.23862769849947414542781583428, −6.86742660467586517402490813562, −6.05206678782802513343165168908, −4.98125562420580675359416664898, −4.20788208517507034780242389873, −3.61377781699580578675299959950, −2.34190888543849055420685026819, −1.64832760032658913135244748281, −0.27964237340053247737940310787,
0.27964237340053247737940310787, 1.64832760032658913135244748281, 2.34190888543849055420685026819, 3.61377781699580578675299959950, 4.20788208517507034780242389873, 4.98125562420580675359416664898, 6.05206678782802513343165168908, 6.86742660467586517402490813562, 7.23862769849947414542781583428, 8.028106340670456030923206080284
