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
\(L(1)\) |
\(\approx\) |
\(3.461849564\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.461849564\) |
\(L(\frac{3}{2})\) |
|
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
−8.229708880062826244512610517689, −8.107806407724230702169848724245, −6.89677858206849337570493242123, −6.56201587917051014614144666856, −5.23908378441880035264412197221, −4.83954014044841435874383885361, −4.11857067341460528949311028083, −3.14957058692443973291051858023, −2.11006660128168517913660186825, −1.07643003061251412209578338863,
1.07643003061251412209578338863, 2.11006660128168517913660186825, 3.14957058692443973291051858023, 4.11857067341460528949311028083, 4.83954014044841435874383885361, 5.23908378441880035264412197221, 6.56201587917051014614144666856, 6.89677858206849337570493242123, 8.107806407724230702169848724245, 8.229708880062826244512610517689