L(s) = 1 | − 0.768·2-s + 3-s − 1.40·4-s + 3.25·5-s − 0.768·6-s + 7-s + 2.62·8-s + 9-s − 2.49·10-s + 1.06·11-s − 1.40·12-s − 6.23·13-s − 0.768·14-s + 3.25·15-s + 0.804·16-s − 2.53·17-s − 0.768·18-s + 1.54·19-s − 4.58·20-s + 21-s − 0.821·22-s + 4.87·23-s + 2.62·24-s + 5.56·25-s + 4.79·26-s + 27-s − 1.40·28-s + ⋯ |
L(s) = 1 | − 0.543·2-s + 0.577·3-s − 0.704·4-s + 1.45·5-s − 0.313·6-s + 0.377·7-s + 0.926·8-s + 0.333·9-s − 0.790·10-s + 0.322·11-s − 0.406·12-s − 1.72·13-s − 0.205·14-s + 0.839·15-s + 0.201·16-s − 0.614·17-s − 0.181·18-s + 0.354·19-s − 1.02·20-s + 0.218·21-s − 0.175·22-s + 1.01·23-s + 0.534·24-s + 1.11·25-s + 0.939·26-s + 0.192·27-s − 0.266·28-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
\(L(1)\) |
\(\approx\) |
\(2.029224202\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.029224202\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.768T + 2T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 2.53T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 + 2.96T + 29T^{2} \) |
| 31 | \( 1 - 0.605T + 31T^{2} \) |
| 37 | \( 1 - 5.49T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 6.96T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 - 1.33T + 67T^{2} \) |
| 71 | \( 1 - 4.53T + 71T^{2} \) |
| 73 | \( 1 - 3.57T + 73T^{2} \) |
| 79 | \( 1 - 0.524T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 6.80T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.811278490696490613675084010899, −7.68142352482917950283679539717, −7.27700954284222242346071419010, −6.29265294589194569612516556801, −5.24107754132381823813423436921, −4.87921971092622422203979466797, −3.89569348758901478495716959653, −2.59193591084490322421412517057, −2.00675845002224308649386454965, −0.893848142922566935359099914566,
0.893848142922566935359099914566, 2.00675845002224308649386454965, 2.59193591084490322421412517057, 3.89569348758901478495716959653, 4.87921971092622422203979466797, 5.24107754132381823813423436921, 6.29265294589194569612516556801, 7.27700954284222242346071419010, 7.68142352482917950283679539717, 8.811278490696490613675084010899