L(s) = 1 | − 9.54·2-s + 27·3-s − 36.8·4-s + 248.·5-s − 257.·6-s − 1.45e3·7-s + 1.57e3·8-s + 729·9-s − 2.37e3·10-s − 6.46e3·11-s − 994.·12-s + 4.14e3·13-s + 1.38e4·14-s + 6.72e3·15-s − 1.03e4·16-s − 2.58e4·17-s − 6.96e3·18-s + 2.68e4·19-s − 9.16e3·20-s − 3.92e4·21-s + 6.17e4·22-s + 5.88e4·23-s + 4.24e4·24-s − 1.61e4·25-s − 3.95e4·26-s + 1.96e4·27-s + 5.35e4·28-s + ⋯ |
L(s) = 1 | − 0.843·2-s + 0.577·3-s − 0.287·4-s + 0.890·5-s − 0.487·6-s − 1.60·7-s + 1.08·8-s + 0.333·9-s − 0.751·10-s − 1.46·11-s − 0.166·12-s + 0.522·13-s + 1.35·14-s + 0.514·15-s − 0.629·16-s − 1.27·17-s − 0.281·18-s + 0.898·19-s − 0.256·20-s − 0.924·21-s + 1.23·22-s + 1.00·23-s + 0.627·24-s − 0.206·25-s − 0.441·26-s + 0.192·27-s + 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.030927411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.030927411\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 - 2.05e5T \) |
good | 2 | \( 1 + 9.54T + 128T^{2} \) |
| 5 | \( 1 - 248.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.45e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.46e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.14e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.58e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.68e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.88e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.37e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.58e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.01e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.46e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.35e4T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.61e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.37e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.87e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.37e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 9.60e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.21e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.18e7T + 8.07e13T^{2} \) |
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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
−10.88686133444356580053780398657, −9.994849030628605684152572311282, −9.451657459074613867257513909962, −8.632979059953803135349446150776, −7.45826320644479793955532152614, −6.32694449905131090933302014907, −4.98572579857301080168359086632, −3.34802161674116155137835919741, −2.20157614224029634445084867187, −0.59870669636246793189453459867,
0.59870669636246793189453459867, 2.20157614224029634445084867187, 3.34802161674116155137835919741, 4.98572579857301080168359086632, 6.32694449905131090933302014907, 7.45826320644479793955532152614, 8.632979059953803135349446150776, 9.451657459074613867257513909962, 9.994849030628605684152572311282, 10.88686133444356580053780398657