| L(s) = 1 | + 0.343·2-s − 3-s − 1.88·4-s − 1.00·5-s − 0.343·6-s + 7-s − 1.33·8-s + 9-s − 0.346·10-s − 5.60·11-s + 1.88·12-s − 6.26·13-s + 0.343·14-s + 1.00·15-s + 3.30·16-s − 7.76·17-s + 0.343·18-s + 5.12·19-s + 1.89·20-s − 21-s − 1.92·22-s − 4.36·23-s + 1.33·24-s − 3.98·25-s − 2.15·26-s − 27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | + 0.243·2-s − 0.577·3-s − 0.940·4-s − 0.450·5-s − 0.140·6-s + 0.377·7-s − 0.471·8-s + 0.333·9-s − 0.109·10-s − 1.69·11-s + 0.543·12-s − 1.73·13-s + 0.0918·14-s + 0.260·15-s + 0.826·16-s − 1.88·17-s + 0.0810·18-s + 1.17·19-s + 0.423·20-s − 0.218·21-s − 0.410·22-s − 0.910·23-s + 0.272·24-s − 0.797·25-s − 0.422·26-s − 0.192·27-s − 0.355·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1730645578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1730645578\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.343T + 2T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 + 6.26T + 13T^{2} \) |
| 17 | \( 1 + 7.76T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 + 0.373T + 31T^{2} \) |
| 37 | \( 1 - 4.00T + 37T^{2} \) |
| 41 | \( 1 - 0.0294T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + 9.95T + 53T^{2} \) |
| 59 | \( 1 + 9.74T + 59T^{2} \) |
| 61 | \( 1 + 0.441T + 61T^{2} \) |
| 67 | \( 1 - 9.96T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 - 0.590T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 + 9.11T + 97T^{2} \) |
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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
−8.283133564756380456483096962042, −7.63325659967855833442107825116, −7.25635543419750843614048862720, −5.88646589770950129589125602437, −5.39769785523887175398735571603, −4.60413494612066206052246433253, −4.27050335698330629696846145573, −2.97435685296223832001873945778, −2.06589097668531134608586703013, −0.22252599508159022884876645273,
0.22252599508159022884876645273, 2.06589097668531134608586703013, 2.97435685296223832001873945778, 4.27050335698330629696846145573, 4.60413494612066206052246433253, 5.39769785523887175398735571603, 5.88646589770950129589125602437, 7.25635543419750843614048862720, 7.63325659967855833442107825116, 8.283133564756380456483096962042
