| L(s) = 1 | + 0.529·2-s − 3-s − 1.71·4-s − 1.77·5-s − 0.529·6-s + 7-s − 1.96·8-s + 9-s − 0.941·10-s + 6.49·11-s + 1.71·12-s + 0.529·14-s + 1.77·15-s + 2.39·16-s − 2.94·17-s + 0.529·18-s + 4.83·19-s + 3.05·20-s − 21-s + 3.43·22-s − 5.77·23-s + 1.96·24-s − 1.83·25-s − 27-s − 1.71·28-s − 2.83·29-s + 0.941·30-s + ⋯ |
| L(s) = 1 | + 0.374·2-s − 0.577·3-s − 0.859·4-s − 0.795·5-s − 0.216·6-s + 0.377·7-s − 0.696·8-s + 0.333·9-s − 0.297·10-s + 1.95·11-s + 0.496·12-s + 0.141·14-s + 0.459·15-s + 0.599·16-s − 0.713·17-s + 0.124·18-s + 1.10·19-s + 0.683·20-s − 0.218·21-s + 0.733·22-s − 1.20·23-s + 0.401·24-s − 0.367·25-s − 0.192·27-s − 0.325·28-s − 0.526·29-s + 0.171·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.169655375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169655375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.529T + 2T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 11 | \( 1 - 6.49T + 11T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 + 5.77T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 + 9.55T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 - 6.39T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 1.28T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 - 7.66T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.764870508235354883395520507179, −7.71639342243540541338587772892, −7.11075390237663780609044862465, −6.15033881035853660856639123554, −5.52797590333693566233282722083, −4.61660166986362019261862130228, −3.89343117721360334667801375739, −3.61359690317342589812879793717, −1.82607774419445572838067052567, −0.63914304032631705766319181516,
0.63914304032631705766319181516, 1.82607774419445572838067052567, 3.61359690317342589812879793717, 3.89343117721360334667801375739, 4.61660166986362019261862130228, 5.52797590333693566233282722083, 6.15033881035853660856639123554, 7.11075390237663780609044862465, 7.71639342243540541338587772892, 8.764870508235354883395520507179
