| L(s) = 1 | − 0.999·2-s + 0.560·3-s − 1.00·4-s + 0.738·5-s − 0.560·6-s − 0.683·7-s + 2.99·8-s − 2.68·9-s − 0.737·10-s − 11-s − 0.561·12-s + 3.37·13-s + 0.682·14-s + 0.413·15-s − 0.993·16-s + 1.72·17-s + 2.68·18-s − 0.455·19-s − 0.739·20-s − 0.383·21-s + 0.999·22-s + 2.24·23-s + 1.68·24-s − 4.45·25-s − 3.37·26-s − 3.18·27-s + 0.684·28-s + ⋯ |
| L(s) = 1 | − 0.706·2-s + 0.323·3-s − 0.500·4-s + 0.330·5-s − 0.228·6-s − 0.258·7-s + 1.06·8-s − 0.895·9-s − 0.233·10-s − 0.301·11-s − 0.162·12-s + 0.937·13-s + 0.182·14-s + 0.106·15-s − 0.248·16-s + 0.417·17-s + 0.632·18-s − 0.104·19-s − 0.165·20-s − 0.0836·21-s + 0.213·22-s + 0.467·23-s + 0.343·24-s − 0.891·25-s − 0.662·26-s − 0.613·27-s + 0.129·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.023428863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023428863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 0.999T + 2T^{2} \) |
| 3 | \( 1 - 0.560T + 3T^{2} \) |
| 5 | \( 1 - 0.738T + 5T^{2} \) |
| 7 | \( 1 + 0.683T + 7T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 + 0.455T + 19T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 - 0.832T + 31T^{2} \) |
| 37 | \( 1 - 9.96T + 37T^{2} \) |
| 41 | \( 1 - 8.80T + 41T^{2} \) |
| 43 | \( 1 - 6.18T + 43T^{2} \) |
| 47 | \( 1 - 7.59T + 47T^{2} \) |
| 53 | \( 1 + 3.68T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 - 6.97T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 + 2.50T + 73T^{2} \) |
| 79 | \( 1 + 1.91T + 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−9.365930284113827758749125451849, −8.863612089580886164614168406556, −8.039735925824223896195666193218, −7.49577286390620411713466579990, −6.12961789387676004776716807960, −5.56607641470111445433026484655, −4.36301681086719018957948597006, −3.41373852036189806562918203823, −2.23847242797317450935244821856, −0.803531227316496143850475220122,
0.803531227316496143850475220122, 2.23847242797317450935244821856, 3.41373852036189806562918203823, 4.36301681086719018957948597006, 5.56607641470111445433026484655, 6.12961789387676004776716807960, 7.49577286390620411713466579990, 8.039735925824223896195666193218, 8.863612089580886164614168406556, 9.365930284113827758749125451849
