| L(s) = 1 | + 1.27·2-s − 0.370·4-s + 4.09·5-s − 7-s − 3.02·8-s + 5.22·10-s + 4.39·13-s − 1.27·14-s − 3.12·16-s + 4.19·17-s + 1.24·19-s − 1.51·20-s − 4.97·23-s + 11.7·25-s + 5.60·26-s + 0.370·28-s − 1.93·29-s − 1.56·31-s + 2.06·32-s + 5.35·34-s − 4.09·35-s − 0.716·37-s + 1.59·38-s − 12.3·40-s + 4.80·41-s − 1.35·43-s − 6.34·46-s + ⋯ |
| L(s) = 1 | + 0.902·2-s − 0.185·4-s + 1.82·5-s − 0.377·7-s − 1.06·8-s + 1.65·10-s + 1.21·13-s − 0.341·14-s − 0.780·16-s + 1.01·17-s + 0.286·19-s − 0.338·20-s − 1.03·23-s + 2.34·25-s + 1.09·26-s + 0.0699·28-s − 0.359·29-s − 0.280·31-s + 0.365·32-s + 0.917·34-s − 0.691·35-s − 0.117·37-s + 0.258·38-s − 1.95·40-s + 0.750·41-s − 0.206·43-s − 0.935·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.241910576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.241910576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + 0.716T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.97T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 7.59T + 67T^{2} \) |
| 71 | \( 1 + 0.218T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 4.56T + 79T^{2} \) |
| 83 | \( 1 - 2.45T + 83T^{2} \) |
| 89 | \( 1 + 4.20T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.87146843628894875955724315122, −6.81518195995717156163345304693, −6.13070765030645662478224775476, −5.70183509031747458086082180300, −5.36798721573090461142230566425, −4.31031046913344776729642165010, −3.56757434994658339672687674417, −2.83151320395344825631447156038, −1.94349687111168461670404572905, −0.936841107284967111840082149347,
0.936841107284967111840082149347, 1.94349687111168461670404572905, 2.83151320395344825631447156038, 3.56757434994658339672687674417, 4.31031046913344776729642165010, 5.36798721573090461142230566425, 5.70183509031747458086082180300, 6.13070765030645662478224775476, 6.81518195995717156163345304693, 7.87146843628894875955724315122
