| L(s) = 1 | − 2.60·2-s − 3-s + 4.77·4-s − 2.58·5-s + 2.60·6-s + 7-s − 7.21·8-s + 9-s + 6.72·10-s − 0.115·11-s − 4.77·12-s + 0.834·13-s − 2.60·14-s + 2.58·15-s + 9.22·16-s − 5.62·17-s − 2.60·18-s + 3.68·19-s − 12.3·20-s − 21-s + 0.301·22-s + 4.67·23-s + 7.21·24-s + 1.68·25-s − 2.17·26-s − 27-s + 4.77·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.577·3-s + 2.38·4-s − 1.15·5-s + 1.06·6-s + 0.377·7-s − 2.55·8-s + 0.333·9-s + 2.12·10-s − 0.0349·11-s − 1.37·12-s + 0.231·13-s − 0.695·14-s + 0.667·15-s + 2.30·16-s − 1.36·17-s − 0.613·18-s + 0.845·19-s − 2.75·20-s − 0.218·21-s + 0.0643·22-s + 0.974·23-s + 1.47·24-s + 0.336·25-s − 0.426·26-s − 0.192·27-s + 0.901·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3527564926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3527564926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 11 | \( 1 + 0.115T + 11T^{2} \) |
| 13 | \( 1 - 0.834T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 - 4.67T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 0.00633T + 43T^{2} \) |
| 47 | \( 1 + 0.636T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 + 8.43T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 5.15T + 67T^{2} \) |
| 71 | \( 1 - 0.734T + 71T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 + 1.05T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.465740199559124127361765021825, −7.80854756033603976613044487766, −7.17776849917223082693139201942, −6.73405635100129422883708329323, −5.72445641060657333415547579051, −4.70903338903996798887188401814, −3.71447494643076190869545873501, −2.60666717130296761798262760105, −1.48856755843143007292319476085, −0.48310616178960200343273759849,
0.48310616178960200343273759849, 1.48856755843143007292319476085, 2.60666717130296761798262760105, 3.71447494643076190869545873501, 4.70903338903996798887188401814, 5.72445641060657333415547579051, 6.73405635100129422883708329323, 7.17776849917223082693139201942, 7.80854756033603976613044487766, 8.465740199559124127361765021825
