| L(s) = 1 | − 20.6·2-s − 27·3-s + 297.·4-s + 165.·5-s + 556.·6-s − 1.38e3·7-s − 3.48e3·8-s + 729·9-s − 3.41e3·10-s + 2.50e3·11-s − 8.02e3·12-s + 5.80e3·13-s + 2.85e4·14-s − 4.46e3·15-s + 3.38e4·16-s − 1.30e4·17-s − 1.50e4·18-s + 1.23e4·19-s + 4.91e4·20-s + 3.74e4·21-s − 5.15e4·22-s − 8.09e4·23-s + 9.41e4·24-s − 5.07e4·25-s − 1.19e5·26-s − 1.96e4·27-s − 4.11e5·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.32·4-s + 0.591·5-s + 1.05·6-s − 1.52·7-s − 2.40·8-s + 0.333·9-s − 1.07·10-s + 0.566·11-s − 1.34·12-s + 0.733·13-s + 2.78·14-s − 0.341·15-s + 2.06·16-s − 0.643·17-s − 0.607·18-s + 0.411·19-s + 1.37·20-s + 0.881·21-s − 1.03·22-s − 1.38·23-s + 1.39·24-s − 0.649·25-s − 1.33·26-s − 0.192·27-s − 3.54·28-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
| \(L(4)\) |
\(\approx\) |
\(0.4133204550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4133204550\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 20.6T + 128T^{2} \) |
| 5 | \( 1 - 165.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.38e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.50e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.80e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.30e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.09e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.82e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.27e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.08e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.87e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.85e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.21e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.92e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.90e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.18e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.02e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.02e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.19e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.45e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.09e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.05e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.683816509420839359790959530559, −9.316002271557741409380045154131, −8.255512129171052690158166617033, −7.14237315806729154364517809354, −6.31045043345200924979342282923, −5.93312411318573097391539038540, −3.85164000840109341409272447614, −2.53030016280472325623203672960, −1.45100358778974055579294906905, −0.40370665431017643446882398211,
0.40370665431017643446882398211, 1.45100358778974055579294906905, 2.53030016280472325623203672960, 3.85164000840109341409272447614, 5.93312411318573097391539038540, 6.31045043345200924979342282923, 7.14237315806729154364517809354, 8.255512129171052690158166617033, 9.316002271557741409380045154131, 9.683816509420839359790959530559
