| L(s) = 1 | + 1.67·2-s − 509.·4-s + 1.53e3·5-s + 5.65e3·7-s − 1.71e3·8-s + 2.58e3·10-s − 1.98e4·11-s + 9.97e4·13-s + 9.50e3·14-s + 2.57e5·16-s − 5.03e5·17-s − 6.70e5·19-s − 7.83e5·20-s − 3.32e4·22-s + 1.93e6·23-s + 4.14e5·25-s + 1.67e5·26-s − 2.88e6·28-s + 4.32e6·29-s + 1.02e6·31-s + 1.31e6·32-s − 8.46e5·34-s + 8.70e6·35-s + 1.81e7·37-s − 1.12e6·38-s − 2.63e6·40-s − 9.30e6·41-s + ⋯ |
| L(s) = 1 | + 0.0742·2-s − 0.994·4-s + 1.10·5-s + 0.890·7-s − 0.148·8-s + 0.0817·10-s − 0.408·11-s + 0.969·13-s + 0.0660·14-s + 0.983·16-s − 1.46·17-s − 1.18·19-s − 1.09·20-s − 0.0302·22-s + 1.43·23-s + 0.212·25-s + 0.0719·26-s − 0.885·28-s + 1.13·29-s + 0.199·31-s + 0.221·32-s − 0.108·34-s + 0.980·35-s + 1.59·37-s − 0.0876·38-s − 0.162·40-s − 0.514·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.255208248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.255208248\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.67T + 512T^{2} \) |
| 5 | \( 1 - 1.53e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.65e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.98e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 9.97e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.03e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.93e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.32e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.02e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.81e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 9.30e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.57e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.83e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.39e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.61e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.45e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.15e6T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.42e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.11e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.71e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.98e8T + 7.60e17T^{2} \) |
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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
−12.99021187473819810857203045260, −11.23722749033001883315590488935, −10.27334718194111777407804504105, −9.000442947870040350527078832973, −8.331261856501678333989935601034, −6.47726004582235483029647290709, −5.26065825893916907497908057518, −4.25205122071406077222340301744, −2.33396761687780766088022054079, −0.913182248682956404973070957075,
0.913182248682956404973070957075, 2.33396761687780766088022054079, 4.25205122071406077222340301744, 5.26065825893916907497908057518, 6.47726004582235483029647290709, 8.331261856501678333989935601034, 9.000442947870040350527078832973, 10.27334718194111777407804504105, 11.23722749033001883315590488935, 12.99021187473819810857203045260
