| L(s) = 1 | − 2-s + 4-s + 3.51·5-s + 2.87·7-s − 8-s − 3.51·10-s − 5.20·11-s + 13-s − 2.87·14-s + 16-s + 3.58·17-s + 3.01·19-s + 3.51·20-s + 5.20·22-s + 7.20·23-s + 7.33·25-s − 26-s + 2.87·28-s + 29-s + 7.94·31-s − 32-s − 3.58·34-s + 10.0·35-s − 3.74·37-s − 3.01·38-s − 3.51·40-s + 0.915·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.57·5-s + 1.08·7-s − 0.353·8-s − 1.11·10-s − 1.56·11-s + 0.277·13-s − 0.767·14-s + 0.250·16-s + 0.868·17-s + 0.692·19-s + 0.785·20-s + 1.10·22-s + 1.50·23-s + 1.46·25-s − 0.196·26-s + 0.542·28-s + 0.185·29-s + 1.42·31-s − 0.176·32-s − 0.614·34-s + 1.70·35-s − 0.615·37-s − 0.489·38-s − 0.555·40-s + 0.142·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.455844490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455844490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 3.01T + 19T^{2} \) |
| 23 | \( 1 - 7.20T + 23T^{2} \) |
| 31 | \( 1 - 7.94T + 31T^{2} \) |
| 37 | \( 1 + 3.74T + 37T^{2} \) |
| 41 | \( 1 - 0.915T + 41T^{2} \) |
| 43 | \( 1 - 5.58T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 2.00T + 67T^{2} \) |
| 71 | \( 1 - 5.62T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.051720711101922050583735710811, −7.44370409159013397562239434809, −6.60971664422503862338008111847, −5.80469233514930006823473726848, −5.20117587544011021602871929246, −4.80124914002075555800224261906, −3.14777862262096093587456016880, −2.57770680554023090429140989576, −1.67569107183628138135868075125, −0.963680716583427686529168385480,
0.963680716583427686529168385480, 1.67569107183628138135868075125, 2.57770680554023090429140989576, 3.14777862262096093587456016880, 4.80124914002075555800224261906, 5.20117587544011021602871929246, 5.80469233514930006823473726848, 6.60971664422503862338008111847, 7.44370409159013397562239434809, 8.051720711101922050583735710811
