| L(s) = 1 | + 2.10·2-s + 3-s + 2.43·4-s + 1.18·5-s + 2.10·6-s + 7-s + 0.913·8-s + 9-s + 2.49·10-s + 4.06·11-s + 2.43·12-s + 2.43·13-s + 2.10·14-s + 1.18·15-s − 2.94·16-s − 3.16·17-s + 2.10·18-s + 6.15·19-s + 2.87·20-s + 21-s + 8.56·22-s + 6.53·23-s + 0.913·24-s − 3.60·25-s + 5.13·26-s + 27-s + 2.43·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 0.577·3-s + 1.21·4-s + 0.529·5-s + 0.859·6-s + 0.377·7-s + 0.322·8-s + 0.333·9-s + 0.787·10-s + 1.22·11-s + 0.702·12-s + 0.676·13-s + 0.562·14-s + 0.305·15-s − 0.736·16-s − 0.767·17-s + 0.496·18-s + 1.41·19-s + 0.643·20-s + 0.218·21-s + 1.82·22-s + 1.36·23-s + 0.186·24-s − 0.720·25-s + 1.00·26-s + 0.192·27-s + 0.459·28-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
| \(L(1)\) |
\(\approx\) |
\(6.843447543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.843447543\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.10T + 2T^{2} \) |
| 5 | \( 1 - 1.18T + 5T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 6.15T + 19T^{2} \) |
| 23 | \( 1 - 6.53T + 23T^{2} \) |
| 29 | \( 1 + 1.70T + 29T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 + 2.49T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 + 9.76T + 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 - 3.85T + 59T^{2} \) |
| 61 | \( 1 - 9.27T + 61T^{2} \) |
| 67 | \( 1 + 3.42T + 67T^{2} \) |
| 71 | \( 1 + 5.60T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 - 4.15T + 79T^{2} \) |
| 83 | \( 1 + 1.48T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.612600800922636610312004229388, −7.37275840380467898688786174590, −6.85014498974611485689573231373, −6.06079995086175679726998398465, −5.35292859436854442364405934571, −4.65942735754272316100233075558, −3.73477211733277764770572472441, −3.32749760919881218503129449259, −2.21161272858635080013217691933, −1.35498463306336250424243849188,
1.35498463306336250424243849188, 2.21161272858635080013217691933, 3.32749760919881218503129449259, 3.73477211733277764770572472441, 4.65942735754272316100233075558, 5.35292859436854442364405934571, 6.06079995086175679726998398465, 6.85014498974611485689573231373, 7.37275840380467898688786174590, 8.612600800922636610312004229388
