| L(s) = 1 | − 2-s + 0.451·3-s + 4-s + 0.275·5-s − 0.451·6-s + 4.58·7-s − 8-s − 2.79·9-s − 0.275·10-s − 4.42·11-s + 0.451·12-s − 6.63·13-s − 4.58·14-s + 0.124·15-s + 16-s − 1.06·17-s + 2.79·18-s + 19-s + 0.275·20-s + 2.07·21-s + 4.42·22-s + 8.81·23-s − 0.451·24-s − 4.92·25-s + 6.63·26-s − 2.61·27-s + 4.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.260·3-s + 0.5·4-s + 0.123·5-s − 0.184·6-s + 1.73·7-s − 0.353·8-s − 0.932·9-s − 0.0872·10-s − 1.33·11-s + 0.130·12-s − 1.84·13-s − 1.22·14-s + 0.0321·15-s + 0.250·16-s − 0.258·17-s + 0.659·18-s + 0.229·19-s + 0.0617·20-s + 0.451·21-s + 0.943·22-s + 1.83·23-s − 0.0921·24-s − 0.984·25-s + 1.30·26-s − 0.503·27-s + 0.866·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.345510827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345510827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 0.451T + 3T^{2} \) |
| 5 | \( 1 - 0.275T + 5T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 + 4.42T + 11T^{2} \) |
| 13 | \( 1 + 6.63T + 13T^{2} \) |
| 17 | \( 1 + 1.06T + 17T^{2} \) |
| 23 | \( 1 - 8.81T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 + 0.241T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 - 3.81T + 67T^{2} \) |
| 71 | \( 1 - 1.40T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 2.74T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 6.86T + 89T^{2} \) |
| 97 | \( 1 - 5.66T + 97T^{2} \) |
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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
−8.009261979856653391772936245413, −7.38842425158368746733440613857, −6.73476511044309258841715858394, −5.51749674914682669446286966680, −5.06629394318265674645148104453, −4.62756210216885040795292233163, −3.09641656545921885554676634369, −2.50774190207992699436789989729, −1.88592726887996977012368270327, −0.60775609486515975834987711764,
0.60775609486515975834987711764, 1.88592726887996977012368270327, 2.50774190207992699436789989729, 3.09641656545921885554676634369, 4.62756210216885040795292233163, 5.06629394318265674645148104453, 5.51749674914682669446286966680, 6.73476511044309258841715858394, 7.38842425158368746733440613857, 8.009261979856653391772936245413
