| L(s) = 1 | + 8·2-s − 105·5-s + 937·7-s − 512·8-s − 840·10-s − 5.94e3·11-s − 68·13-s + 7.49e3·14-s − 4.09e3·16-s − 1.08e4·17-s − 9.67e4·19-s − 4.75e4·22-s + 642·23-s + 7.81e4·25-s − 544·26-s + 1.25e5·29-s + 1.61e5·31-s − 8.64e4·34-s − 9.83e4·35-s − 8.28e5·37-s − 7.74e5·38-s + 5.37e4·40-s + 6.27e5·41-s − 5.70e5·43-s + 5.13e3·46-s − 5.38e5·47-s + 8.23e5·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.375·5-s + 1.03·7-s − 0.353·8-s − 0.265·10-s − 1.34·11-s − 0.00858·13-s + 0.730·14-s − 1/4·16-s − 0.533·17-s − 3.23·19-s − 0.951·22-s + 0.0110·23-s + 25-s − 0.00607·26-s + 0.958·29-s + 0.972·31-s − 0.376·34-s − 0.387·35-s − 2.68·37-s − 2.28·38-s + 0.132·40-s + 1.42·41-s − 1.09·43-s + 0.00777·46-s − 0.756·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2566485318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2566485318\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 21 p T - 2684 p^{2} T^{2} + 21 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 937 T + 54426 T^{2} - 937 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5943 T + 15832078 T^{2} + 5943 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 68 T - 62743893 T^{2} + 68 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5400 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 48382 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 642 T - 3404413283 T^{2} - 642 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 125934 T - 1390503953 T^{2} - 125934 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 161275 T - 1502988486 T^{2} - 161275 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 414286 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 627474 T + 198969346795 T^{2} - 627474 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 570590 T + 53754336993 T^{2} + 570590 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 538698 T - 216427585259 T^{2} + 538698 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 356283 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 2910828 T + 5984268160765 T^{2} - 2910828 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2684168 T + 4062015016203 T^{2} + 2684168 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2681078 T + 1127467636761 T^{2} + 2681078 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3705480 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 153151 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 7579288 T + 38241697600785 T^{2} - 7579288 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9345999 T + 60211646318374 T^{2} + 9345999 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4033602 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 5754097 T - 47688652192704 T^{2} - 5754097 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.86803780532767277880916977193, −11.36081642647129327690491127999, −10.63922575693065816701557271678, −10.50839813818373489793781086757, −10.17376562695668721424968254310, −8.785636667636137869825710456962, −8.732660950638973057949848746797, −8.352736408486258643131642861354, −7.67828350705093600110579970973, −7.00634123997393663251579300205, −6.43786513984403119045272190692, −5.89602343016111180598322192977, −4.98318360377482053072611225905, −4.74576157504504864629955589374, −4.29836610279160154508257187595, −3.51671986314561326905454866246, −2.57238236659697114690109611032, −2.23025041889363898355824158920, −1.31117764776791026005286469036, −0.11721944378487675486094117467,
0.11721944378487675486094117467, 1.31117764776791026005286469036, 2.23025041889363898355824158920, 2.57238236659697114690109611032, 3.51671986314561326905454866246, 4.29836610279160154508257187595, 4.74576157504504864629955589374, 4.98318360377482053072611225905, 5.89602343016111180598322192977, 6.43786513984403119045272190692, 7.00634123997393663251579300205, 7.67828350705093600110579970973, 8.352736408486258643131642861354, 8.732660950638973057949848746797, 8.785636667636137869825710456962, 10.17376562695668721424968254310, 10.50839813818373489793781086757, 10.63922575693065816701557271678, 11.36081642647129327690491127999, 11.86803780532767277880916977193
