| L(s) = 1 | + 6.55e4·4-s − 1.24e6·7-s − 1.43e7·9-s + 3.22e9·16-s − 6.10e10·25-s − 8.15e10·28-s − 9.40e11·36-s − 2.18e12·37-s − 2.88e12·43-s − 3.19e12·49-s + 1.78e13·63-s + 1.40e14·64-s + 1.98e14·67-s − 1.77e14·79-s + 2.05e14·81-s − 4.00e15·100-s + 7.56e15·109-s − 4.01e15·112-s + 8.35e15·121-s + 127-s + 131-s + 137-s + 139-s − 4.62e16·144-s − 1.42e17·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·4-s − 0.571·7-s − 9-s + 3·16-s − 2·25-s − 1.14·28-s − 2·36-s − 3.77·37-s − 1.61·43-s − 0.673·49-s + 0.571·63-s + 4·64-s + 3.99·67-s − 1.03·79-s + 81-s − 4·100-s + 3.96·109-s − 1.71·112-s + 2·121-s − 3·144-s − 7.55·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.749166253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.749166253\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{15} T^{2} \) |
| 7 | $C_2$ | \( 1 + 1244900 T + p^{15} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 397771850 T + p^{15} T^{2} )( 1 + 397771850 T + p^{15} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7700827736 T + p^{15} T^{2} )( 1 + 7700827736 T + p^{15} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 213681227452 T + p^{15} T^{2} )( 1 + 213681227452 T + p^{15} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 1090158909950 T + p^{15} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 1440654152600 T + p^{15} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 40241378988902 T + p^{15} T^{2} )( 1 + 40241378988902 T + p^{15} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 99059017336400 T + p^{15} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9014812804550 T + p^{15} T^{2} )( 1 + 9014812804550 T + p^{15} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 88692309079036 T + p^{15} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1035097921427150 T + p^{15} T^{2} )( 1 + 1035097921427150 T + p^{15} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.49324509119703693504009919440, −14.24082859765475427127382918343, −13.82914797863283444878026347711, −12.68591442736919373644606412592, −12.18309682557366742813087522021, −11.47215887844121575066050454796, −11.24357202339979074206706482467, −10.22921667753242064975918382585, −9.895183044066066667924107343752, −8.594137534394891621487304573129, −8.005847599001367050501114729258, −7.06305225098507092317833600335, −6.63712617479463320711074826739, −5.84155133974696674709532997182, −5.28425699911513668261171264844, −3.42968457279867998317305634211, −3.33454189204525733152853180207, −2.07261344498227344383184156871, −1.84438923898225316045784140169, −0.46919766004656336527176456339,
0.46919766004656336527176456339, 1.84438923898225316045784140169, 2.07261344498227344383184156871, 3.33454189204525733152853180207, 3.42968457279867998317305634211, 5.28425699911513668261171264844, 5.84155133974696674709532997182, 6.63712617479463320711074826739, 7.06305225098507092317833600335, 8.005847599001367050501114729258, 8.594137534394891621487304573129, 9.895183044066066667924107343752, 10.22921667753242064975918382585, 11.24357202339979074206706482467, 11.47215887844121575066050454796, 12.18309682557366742813087522021, 12.68591442736919373644606412592, 13.82914797863283444878026347711, 14.24082859765475427127382918343, 15.49324509119703693504009919440
