L(s) = 1 | − 2-s + 4-s + 5-s + 3.68·7-s − 8-s − 10-s − 2.14·11-s − 1.48·13-s − 3.68·14-s + 16-s + 0.707·17-s − 1.48·19-s + 20-s + 2.14·22-s − 5.88·23-s + 25-s + 1.48·26-s + 3.68·28-s − 1.80·29-s + 10.0·31-s − 32-s − 0.707·34-s + 3.68·35-s − 37-s + 1.48·38-s − 40-s + 9.00·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.39·7-s − 0.353·8-s − 0.316·10-s − 0.647·11-s − 0.413·13-s − 0.985·14-s + 0.250·16-s + 0.171·17-s − 0.341·19-s + 0.223·20-s + 0.457·22-s − 1.22·23-s + 0.200·25-s + 0.292·26-s + 0.696·28-s − 0.334·29-s + 1.80·31-s − 0.176·32-s − 0.121·34-s + 0.623·35-s − 0.164·37-s + 0.241·38-s − 0.158·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.631642508\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.631642508\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 3.68T + 7T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 0.707T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 5.88T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 41 | \( 1 - 9.00T + 41T^{2} \) |
| 43 | \( 1 - 2.36T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 3.63T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 7.66T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 - 4.46T + 89T^{2} \) |
| 97 | \( 1 + 7.00T + 97T^{2} \) |
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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
−8.444760116329346983115807190918, −7.990642156677898568815920299509, −7.43874242584827259693830293058, −6.41231783032005049966169834712, −5.64545064669141237040860683126, −4.88565833905982814101331461704, −4.04001006204951191029478885204, −2.58596762849953509123924390256, −2.02642750116839198045490755436, −0.861145527082407018024994211270,
0.861145527082407018024994211270, 2.02642750116839198045490755436, 2.58596762849953509123924390256, 4.04001006204951191029478885204, 4.88565833905982814101331461704, 5.64545064669141237040860683126, 6.41231783032005049966169834712, 7.43874242584827259693830293058, 7.990642156677898568815920299509, 8.444760116329346983115807190918