| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 7-s − 3·8-s + 9-s − 8·11-s + 12-s + 4·13-s − 14-s − 16-s − 12·17-s + 18-s − 8·19-s + 21-s − 8·22-s + 3·24-s − 6·25-s + 4·26-s − 27-s + 28-s + 4·29-s + 5·32-s + 8·33-s − 12·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 2.41·11-s + 0.288·12-s + 1.10·13-s − 0.267·14-s − 1/4·16-s − 2.91·17-s + 0.235·18-s − 1.83·19-s + 0.218·21-s − 1.70·22-s + 0.612·24-s − 6/5·25-s + 0.784·26-s − 0.192·27-s + 0.188·28-s + 0.742·29-s + 0.883·32-s + 1.39·33-s − 2.05·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.123916451859490481483222873179, −7.48171223884197946068743795147, −6.81422856027217448872168795499, −6.47604188152629865155494776849, −6.00140300914464627775014931904, −5.73267711633132091527857634664, −5.08548839023881336508605977410, −4.61914309368534933978095799917, −4.19315915887555872024919481174, −3.94328917656394473634657471263, −2.78781035361267366934893473837, −2.64839746138688614832708523229, −1.80984559120091606425042866771, 0, 0,
1.80984559120091606425042866771, 2.64839746138688614832708523229, 2.78781035361267366934893473837, 3.94328917656394473634657471263, 4.19315915887555872024919481174, 4.61914309368534933978095799917, 5.08548839023881336508605977410, 5.73267711633132091527857634664, 6.00140300914464627775014931904, 6.47604188152629865155494776849, 6.81422856027217448872168795499, 7.48171223884197946068743795147, 8.123916451859490481483222873179
