| L(s) = 1 | − 8.95·2-s − 27·3-s − 47.7·4-s + 476.·5-s + 241.·6-s + 984.·7-s + 1.57e3·8-s + 729·9-s − 4.27e3·10-s − 2.91e3·11-s + 1.28e3·12-s − 1.43e3·13-s − 8.81e3·14-s − 1.28e4·15-s − 7.98e3·16-s + 2.01e4·17-s − 6.53e3·18-s + 1.86e4·19-s − 2.27e4·20-s − 2.65e4·21-s + 2.61e4·22-s + 3.26e3·23-s − 4.25e4·24-s + 1.49e5·25-s + 1.28e4·26-s − 1.96e4·27-s − 4.70e4·28-s + ⋯ |
| L(s) = 1 | − 0.791·2-s − 0.577·3-s − 0.373·4-s + 1.70·5-s + 0.457·6-s + 1.08·7-s + 1.08·8-s + 0.333·9-s − 1.35·10-s − 0.661·11-s + 0.215·12-s − 0.180·13-s − 0.858·14-s − 0.984·15-s − 0.487·16-s + 0.995·17-s − 0.263·18-s + 0.624·19-s − 0.636·20-s − 0.626·21-s + 0.523·22-s + 0.0559·23-s − 0.627·24-s + 1.90·25-s + 0.143·26-s − 0.192·27-s − 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.009679157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009679157\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 + 8.95T + 128T^{2} \) |
| 5 | \( 1 - 476.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 984.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.91e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.43e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.01e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.86e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.26e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.42e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.62e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.17e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.69e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.13e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.81e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.19e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.53e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.69e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.66e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.39e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.73e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.36e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.69e7T + 8.07e13T^{2} \) |
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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
−9.892985780282578683458450961694, −9.180150127310108799043656838804, −8.132734403770918753383286072101, −7.34079228380448580938500272259, −5.96571615510148947781172331060, −5.27672127456420669391446733752, −4.53990123372424779623922467722, −2.59065736338052759777499862583, −1.46566551238772189126763922195, −0.838419363357476595449531132157,
0.838419363357476595449531132157, 1.46566551238772189126763922195, 2.59065736338052759777499862583, 4.53990123372424779623922467722, 5.27672127456420669391446733752, 5.96571615510148947781172331060, 7.34079228380448580938500272259, 8.132734403770918753383286072101, 9.180150127310108799043656838804, 9.892985780282578683458450961694
