| L(s) = 1 | + 32·2-s − 265.·3-s + 1.02e3·4-s + 1.58e3·5-s − 8.50e3·6-s + 1.76e4·7-s + 3.27e4·8-s − 1.06e5·9-s + 5.08e4·10-s + 2.60e5·11-s − 2.72e5·12-s + 3.71e5·13-s + 5.65e5·14-s − 4.22e5·15-s + 1.04e6·16-s + 1.15e7·17-s − 3.40e6·18-s + 1.63e7·19-s + 1.62e6·20-s − 4.70e6·21-s + 8.33e6·22-s + 2.26e6·23-s − 8.71e6·24-s − 4.63e7·25-s + 1.18e7·26-s + 7.54e7·27-s + 1.81e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.631·3-s + 0.5·4-s + 0.227·5-s − 0.446·6-s + 0.397·7-s + 0.353·8-s − 0.600·9-s + 0.160·10-s + 0.487·11-s − 0.315·12-s + 0.277·13-s + 0.281·14-s − 0.143·15-s + 0.250·16-s + 1.97·17-s − 0.424·18-s + 1.51·19-s + 0.113·20-s − 0.251·21-s + 0.345·22-s + 0.0733·23-s − 0.223·24-s − 0.948·25-s + 0.196·26-s + 1.01·27-s + 0.198·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.707884363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.707884363\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 13 | \( 1 - 3.71e5T \) |
| good | 3 | \( 1 + 265.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 1.58e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 1.76e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.60e5T + 2.85e11T^{2} \) |
| 17 | \( 1 - 1.15e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.63e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.26e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 5.99e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.20e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.53e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.37e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.16e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 4.67e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.04e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 5.94e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.93e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 6.58e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.93e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.81e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.20e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.11e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.40e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.31e11T + 7.15e21T^{2} \) |
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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
−14.62189112058253093170600284386, −13.77416630613446769700594460505, −12.12076150565704319500330551736, −11.42054244300979374127282704006, −9.862034173782571472904934806324, −7.87484962023263746530662692245, −6.13090253011180035440745864964, −5.11402028918428597196209743189, −3.24696759210686600837462189650, −1.16774677072022394489623193085,
1.16774677072022394489623193085, 3.24696759210686600837462189650, 5.11402028918428597196209743189, 6.13090253011180035440745864964, 7.87484962023263746530662692245, 9.862034173782571472904934806324, 11.42054244300979374127282704006, 12.12076150565704319500330551736, 13.77416630613446769700594460505, 14.62189112058253093170600284386
