L(s) = 1 | − 2-s − 2.87·3-s + 4-s − 5-s + 2.87·6-s − 0.638·7-s − 8-s + 5.26·9-s + 10-s − 11-s − 2.87·12-s − 4.59·13-s + 0.638·14-s + 2.87·15-s + 16-s + 2.19·17-s − 5.26·18-s + 8.48·19-s − 20-s + 1.83·21-s + 22-s − 6.59·23-s + 2.87·24-s + 25-s + 4.59·26-s − 6.52·27-s − 0.638·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.66·3-s + 0.5·4-s − 0.447·5-s + 1.17·6-s − 0.241·7-s − 0.353·8-s + 1.75·9-s + 0.316·10-s − 0.301·11-s − 0.830·12-s − 1.27·13-s + 0.170·14-s + 0.742·15-s + 0.250·16-s + 0.533·17-s − 1.24·18-s + 1.94·19-s − 0.223·20-s + 0.400·21-s + 0.213·22-s − 1.37·23-s + 0.586·24-s + 0.200·25-s + 0.901·26-s − 1.25·27-s − 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3125547182\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3125547182\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 + 2.87T + 3T^{2} \) |
| 7 | \( 1 + 0.638T + 7T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 19 | \( 1 - 8.48T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 - 8.68T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 47 | \( 1 + 7.43T + 47T^{2} \) |
| 53 | \( 1 + 1.66T + 53T^{2} \) |
| 59 | \( 1 - 9.08T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 3.07T + 67T^{2} \) |
| 71 | \( 1 + 2.59T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 6.29T + 79T^{2} \) |
| 83 | \( 1 + 2.36T + 83T^{2} \) |
| 89 | \( 1 - 3.41T + 89T^{2} \) |
| 97 | \( 1 - 6.87T + 97T^{2} \) |
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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
−7.88620171266471347377866849792, −7.61271096954075912324945678265, −6.95802355828485560604915658187, −6.04083576640832373816856910261, −5.51225577578168493032724875629, −4.85073595301721171544866092146, −3.86117821083868745042615386064, −2.77088780689194645941888501067, −1.48157224347633364163675740553, −0.38912143377736055310545186550,
0.38912143377736055310545186550, 1.48157224347633364163675740553, 2.77088780689194645941888501067, 3.86117821083868745042615386064, 4.85073595301721171544866092146, 5.51225577578168493032724875629, 6.04083576640832373816856910261, 6.95802355828485560604915658187, 7.61271096954075912324945678265, 7.88620171266471347377866849792