| L(s) = 1 | − 9·2-s + 11·4-s + 18·5-s + 98·7-s + 243·8-s − 162·10-s − 396·11-s − 350·13-s − 882·14-s − 1.38e3·16-s − 1.80e3·17-s − 3.26e3·19-s + 198·20-s + 3.56e3·22-s − 2.08e3·23-s − 4.58e3·25-s + 3.15e3·26-s + 1.07e3·28-s − 6.69e3·29-s − 20·31-s + 1.53e3·32-s + 1.62e4·34-s + 1.76e3·35-s + 6.23e3·37-s + 2.93e4·38-s + 4.37e3·40-s + 6.04e3·41-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 0.343·4-s + 0.321·5-s + 0.755·7-s + 1.34·8-s − 0.512·10-s − 0.986·11-s − 0.574·13-s − 1.20·14-s − 1.35·16-s − 1.51·17-s − 2.07·19-s + 0.110·20-s + 1.56·22-s − 0.823·23-s − 1.46·25-s + 0.913·26-s + 0.259·28-s − 1.47·29-s − 0.00373·31-s + 0.265·32-s + 2.40·34-s + 0.243·35-s + 0.748·37-s + 3.30·38-s + 0.432·40-s + 0.561·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 9 T + 35 p T^{2} + 9 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 18 T + 4906 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 36 p T + 142198 T^{2} + 36 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 350 T + 546978 T^{2} + 350 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1800 T + 3567406 T^{2} + 1800 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3266 T + 7614270 T^{2} + 3266 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2088 T + 9365230 T^{2} + 2088 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6696 T + 51326470 T^{2} + 6696 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 20 T + 53103102 T^{2} + 20 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6232 T + 144242070 T^{2} - 6232 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6048 T + 223864366 T^{2} - 6048 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3020 T - 30383466 T^{2} + 3020 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 11700 T + 292735582 T^{2} + 11700 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9468 T + 858185230 T^{2} + 9468 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 43938 T + 1852599934 T^{2} - 43938 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 64754 T + 2408321418 T^{2} + 64754 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24784 T + 2799959190 T^{2} - 24784 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 97416 T + 5729557966 T^{2} + 97416 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 17452 T + 3828622374 T^{2} - 17452 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 51256 T + 3645565854 T^{2} - 51256 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 117558 T + 7798161502 T^{2} + 117558 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 84276 T + 5915697430 T^{2} + 84276 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20776 T + 16174049358 T^{2} - 20776 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.40068732379135880809024521843, −13.32859028945119021848845057503, −12.69993870607601881754929643237, −11.79235286957854930258046095508, −10.96738464651369409102198206617, −10.83053485270417759958028197589, −9.895454711119492587162477480760, −9.770019056205913469553285644602, −8.786663241086592585910118323794, −8.622324041671088209687833388445, −7.80627524892866774731121675700, −7.55016373329611929872680324318, −6.41975590104036289706913594785, −5.63164621465331210356143094652, −4.58694324331215130232623911361, −4.18210231475701305866080517294, −2.30881229721857315672749637921, −1.76249649689538642059159317241, 0, 0,
1.76249649689538642059159317241, 2.30881229721857315672749637921, 4.18210231475701305866080517294, 4.58694324331215130232623911361, 5.63164621465331210356143094652, 6.41975590104036289706913594785, 7.55016373329611929872680324318, 7.80627524892866774731121675700, 8.622324041671088209687833388445, 8.786663241086592585910118323794, 9.770019056205913469553285644602, 9.895454711119492587162477480760, 10.83053485270417759958028197589, 10.96738464651369409102198206617, 11.79235286957854930258046095508, 12.69993870607601881754929643237, 13.32859028945119021848845057503, 13.40068732379135880809024521843
