| L(s) = 1 | − 2.57·2-s + 2.85·3-s + 4.65·4-s + 3.17·5-s − 7.37·6-s − 3.48·7-s − 6.84·8-s + 5.17·9-s − 8.18·10-s + 11-s + 13.3·12-s + 1.45·13-s + 8.98·14-s + 9.07·15-s + 8.34·16-s + 17-s − 13.3·18-s + 7.42·19-s + 14.7·20-s − 9.95·21-s − 2.57·22-s + 6.93·23-s − 19.5·24-s + 5.06·25-s − 3.75·26-s + 6.20·27-s − 16.2·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 1.65·3-s + 2.32·4-s + 1.41·5-s − 3.01·6-s − 1.31·7-s − 2.42·8-s + 1.72·9-s − 2.58·10-s + 0.301·11-s + 3.84·12-s + 0.403·13-s + 2.40·14-s + 2.34·15-s + 2.08·16-s + 0.242·17-s − 3.14·18-s + 1.70·19-s + 3.30·20-s − 2.17·21-s − 0.549·22-s + 1.44·23-s − 3.99·24-s + 1.01·25-s − 0.736·26-s + 1.19·27-s − 3.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287713320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287713320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 - 2.85T + 3T^{2} \) |
| 5 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 + 3.48T + 7T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 0.00693T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 7.85T + 53T^{2} \) |
| 59 | \( 1 + 6.07T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 - 0.816T + 67T^{2} \) |
| 71 | \( 1 - 9.08T + 71T^{2} \) |
| 73 | \( 1 - 1.53T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 5.34T + 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.939898547175993682270086316629, −7.40635269782324891273213042493, −6.77064908541851380961075768777, −6.17931499673163384944719175045, −5.33628854619707461549218341438, −3.71054919419841161939367610110, −2.97991638435073267616995463695, −2.58227933644707345420306855352, −1.64544827143796390086012696884, −0.968631586072810836091408923887,
0.968631586072810836091408923887, 1.64544827143796390086012696884, 2.58227933644707345420306855352, 2.97991638435073267616995463695, 3.71054919419841161939367610110, 5.33628854619707461549218341438, 6.17931499673163384944719175045, 6.77064908541851380961075768777, 7.40635269782324891273213042493, 7.939898547175993682270086316629
