| L(s) = 1 | − 4.34·2-s − 2.32·3-s + 10.8·4-s − 2.24·5-s + 10.1·6-s − 12.5·8-s − 21.5·9-s + 9.77·10-s − 17.9·11-s − 25.3·12-s + 16.6·13-s + 5.23·15-s − 32.5·16-s − 21.0·17-s + 93.7·18-s − 74.5·19-s − 24.4·20-s + 78.1·22-s − 9.45·23-s + 29.2·24-s − 119.·25-s − 72.4·26-s + 113.·27-s − 146.·29-s − 22.7·30-s − 249.·31-s + 241.·32-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.448·3-s + 1.36·4-s − 0.201·5-s + 0.688·6-s − 0.554·8-s − 0.798·9-s + 0.308·10-s − 0.493·11-s − 0.610·12-s + 0.355·13-s + 0.0901·15-s − 0.508·16-s − 0.299·17-s + 1.22·18-s − 0.899·19-s − 0.273·20-s + 0.757·22-s − 0.0857·23-s + 0.248·24-s − 0.959·25-s − 0.546·26-s + 0.806·27-s − 0.937·29-s − 0.138·30-s − 1.44·31-s + 1.33·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01848801760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01848801760\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 4.34T + 8T^{2} \) |
| 3 | \( 1 + 2.32T + 27T^{2} \) |
| 5 | \( 1 + 2.24T + 125T^{2} \) |
| 11 | \( 1 + 17.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 21.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.45T + 1.21e4T^{2} \) |
| 29 | \( 1 + 146.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 249.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 121.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 434.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 86.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 572.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 84.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 25.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.17e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 957.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 352.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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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
−8.833345100677295604968914969919, −7.88820129930934468648842511911, −7.42870386866984292352122836921, −6.40782593464503436621635462368, −5.77089374160033122977344059071, −4.73794614085636411091531614275, −3.59897674803371605235748647262, −2.39319544548092081714380228728, −1.52624490003326102414728277269, −0.07791586592345623818936032452,
0.07791586592345623818936032452, 1.52624490003326102414728277269, 2.39319544548092081714380228728, 3.59897674803371605235748647262, 4.73794614085636411091531614275, 5.77089374160033122977344059071, 6.40782593464503436621635462368, 7.42870386866984292352122836921, 7.88820129930934468648842511911, 8.833345100677295604968914969919
