L(s) = 1 | − 27.1·2-s − 81·3-s + 226.·4-s − 1.81e3·5-s + 2.20e3·6-s − 9.88e3·7-s + 7.75e3·8-s + 6.56e3·9-s + 4.92e4·10-s + 6.29e4·11-s − 1.83e4·12-s + 8.96e4·13-s + 2.68e5·14-s + 1.46e5·15-s − 3.26e5·16-s − 9.81e4·17-s − 1.78e5·18-s − 6.08e5·19-s − 4.10e5·20-s + 8.00e5·21-s − 1.71e6·22-s − 1.34e6·23-s − 6.28e5·24-s + 1.33e6·25-s − 2.43e6·26-s − 5.31e5·27-s − 2.23e6·28-s + ⋯ |
L(s) = 1 | − 1.20·2-s − 0.577·3-s + 0.442·4-s − 1.29·5-s + 0.693·6-s − 1.55·7-s + 0.669·8-s + 0.333·9-s + 1.55·10-s + 1.29·11-s − 0.255·12-s + 0.870·13-s + 1.86·14-s + 0.748·15-s − 1.24·16-s − 0.285·17-s − 0.400·18-s − 1.07·19-s − 0.573·20-s + 0.898·21-s − 1.55·22-s − 1.00·23-s − 0.386·24-s + 0.681·25-s − 1.04·26-s − 0.192·27-s − 0.688·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.04442201481\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04442201481\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
good | 2 | \( 1 + 27.1T + 512T^{2} \) |
| 5 | \( 1 + 1.81e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.88e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.29e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 8.96e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 9.81e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.08e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.34e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.65e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.43e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.05e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 9.67e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.94e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.14e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.88e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.34e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.50e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.62e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.68e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.09e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.90e7T + 7.60e17T^{2} \) |
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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
−10.88008967422863461203634201524, −9.929968966092691650484728555024, −9.002266757051059542507143283093, −8.168682979610144813581337392080, −6.91065475072410366427124093269, −6.30578151451723988840483739693, −4.29793495965231152195578054391, −3.53988198978244068225714310158, −1.47654749076262994808241357424, −0.13522423139633208937914517139,
0.13522423139633208937914517139, 1.47654749076262994808241357424, 3.53988198978244068225714310158, 4.29793495965231152195578054391, 6.30578151451723988840483739693, 6.91065475072410366427124093269, 8.168682979610144813581337392080, 9.002266757051059542507143283093, 9.929968966092691650484728555024, 10.88008967422863461203634201524