| L(s) = 1 | + 0.759·2-s − 2.25·3-s − 1.42·4-s + 5-s − 1.71·6-s + 4.95·7-s − 2.59·8-s + 2.09·9-s + 0.759·10-s − 11-s + 3.21·12-s + 0.499·13-s + 3.75·14-s − 2.25·15-s + 0.874·16-s − 4.34·17-s + 1.59·18-s − 19-s − 1.42·20-s − 11.1·21-s − 0.759·22-s − 2.78·23-s + 5.86·24-s + 25-s + 0.378·26-s + 2.03·27-s − 7.05·28-s + ⋯ |
| L(s) = 1 | + 0.536·2-s − 1.30·3-s − 0.711·4-s + 0.447·5-s − 0.699·6-s + 1.87·7-s − 0.918·8-s + 0.699·9-s + 0.240·10-s − 0.301·11-s + 0.928·12-s + 0.138·13-s + 1.00·14-s − 0.583·15-s + 0.218·16-s − 1.05·17-s + 0.375·18-s − 0.229·19-s − 0.318·20-s − 2.44·21-s − 0.161·22-s − 0.581·23-s + 1.19·24-s + 0.200·25-s + 0.0743·26-s + 0.391·27-s − 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322726495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322726495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.759T + 2T^{2} \) |
| 3 | \( 1 + 2.25T + 3T^{2} \) |
| 7 | \( 1 - 4.95T + 7T^{2} \) |
| 13 | \( 1 - 0.499T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 - 8.84T + 29T^{2} \) |
| 31 | \( 1 - 1.67T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 2.65T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 2.84T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 - 0.0194T + 97T^{2} \) |
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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
−10.30728938952265820501309085368, −8.918802811314968053009792563308, −8.417989894292391253605665046994, −7.28764444058576914782079793463, −6.04670338689007252659706132710, −5.59621799603212288411435505408, −4.60696061634001472260848610240, −4.39533004064344175898891492634, −2.42239534740499062902560426424, −0.909463286729259167564313350100,
0.909463286729259167564313350100, 2.42239534740499062902560426424, 4.39533004064344175898891492634, 4.60696061634001472260848610240, 5.59621799603212288411435505408, 6.04670338689007252659706132710, 7.28764444058576914782079793463, 8.417989894292391253605665046994, 8.918802811314968053009792563308, 10.30728938952265820501309085368
