| L(s) = 1 | − 8·4-s + 18·7-s − 90·11-s − 65·13-s − 117·17-s − 42·19-s + 18·23-s + 223·25-s − 144·28-s − 99·29-s + 195·37-s + 63·41-s + 82·43-s + 720·44-s − 127·49-s + 520·52-s + 522·53-s + 1.36e3·59-s + 719·61-s + 512·64-s − 1.21e3·67-s + 936·68-s + 810·71-s + 336·76-s − 1.62e3·77-s − 880·79-s − 2.62e3·89-s + ⋯ |
| L(s) = 1 | − 4-s + 0.971·7-s − 2.46·11-s − 1.38·13-s − 1.66·17-s − 0.507·19-s + 0.163·23-s + 1.78·25-s − 0.971·28-s − 0.633·29-s + 0.866·37-s + 0.239·41-s + 0.290·43-s + 2.46·44-s − 0.370·49-s + 1.38·52-s + 1.35·53-s + 3.01·59-s + 1.50·61-s + 64-s − 2.22·67-s + 1.66·68-s + 1.35·71-s + 0.507·76-s − 2.39·77-s − 1.25·79-s − 3.12·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6719251350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6719251350\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 223 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 18 T + 451 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 90 T + 4031 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 117 T + 8776 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 42 T + 7447 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 18 T - 11843 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 99 T - 14588 T^{2} + 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 21950 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 195 T + 63328 T^{2} - 195 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 63 T + 70244 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 82 T - 72783 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 202354 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 261 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 1368 T + 829187 T^{2} - 1368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 901 T + p^{3} T^{2} )( 1 + 182 T + p^{3} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 1218 T + 795271 T^{2} + 1218 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 810 T + 576611 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 309959 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 440 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 284726 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2628 T + 3007097 T^{2} + 2628 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2004 T + 2251345 T^{2} + 2004 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−13.05771874583364804611106677908, −12.95394563306508863529852704814, −12.68064059055099393617718597957, −11.55045472453187082248003071398, −11.27452949265744202545108335743, −10.62127225515077707423305950280, −10.24712723831116510034059204074, −9.627790666267873245272888912027, −8.937311633364286390907241895022, −8.344996066261117028696859587830, −8.188425080815976218225060093168, −7.12709172042544297898385412472, −7.04690718331938845595254147137, −5.62911058220023515251421581793, −5.16005717873446258472430934273, −4.68499724603489142507592209860, −4.22199163034342550816395719015, −2.65474385746469661570119724731, −2.30767924431554354842679496555, −0.42500067284976818809626589803,
0.42500067284976818809626589803, 2.30767924431554354842679496555, 2.65474385746469661570119724731, 4.22199163034342550816395719015, 4.68499724603489142507592209860, 5.16005717873446258472430934273, 5.62911058220023515251421581793, 7.04690718331938845595254147137, 7.12709172042544297898385412472, 8.188425080815976218225060093168, 8.344996066261117028696859587830, 8.937311633364286390907241895022, 9.627790666267873245272888912027, 10.24712723831116510034059204074, 10.62127225515077707423305950280, 11.27452949265744202545108335743, 11.55045472453187082248003071398, 12.68064059055099393617718597957, 12.95394563306508863529852704814, 13.05771874583364804611106677908
