| L(s) = 1 | + 2-s + 2·4-s + 4·5-s + 2·7-s + 5·8-s + 4·10-s + 11-s + 2·13-s + 2·14-s + 5·16-s + 4·17-s − 12·19-s + 8·20-s + 22-s − 4·23-s + 5·25-s + 2·26-s + 4·28-s + 6·29-s − 4·31-s + 10·32-s + 4·34-s + 8·35-s − 12·37-s − 12·38-s + 20·40-s + 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 1.78·5-s + 0.755·7-s + 1.76·8-s + 1.26·10-s + 0.301·11-s + 0.554·13-s + 0.534·14-s + 5/4·16-s + 0.970·17-s − 2.75·19-s + 1.78·20-s + 0.213·22-s − 0.834·23-s + 25-s + 0.392·26-s + 0.755·28-s + 1.11·29-s − 0.718·31-s + 1.76·32-s + 0.685·34-s + 1.35·35-s − 1.97·37-s − 1.94·38-s + 3.16·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.745359457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.745359457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.44075119996207507133698932632, −10.09706287191021982127093132817, −9.734402582732179603870549351118, −8.968122324174631836059259933793, −8.594974867359990464099976452449, −8.371190256571500973490244011435, −7.51418003949278447526066592128, −7.47986182454641841972791260896, −6.64514109159385122075729483782, −6.34102406704256646259256177267, −6.08642031414999280199076545080, −5.50832957864501331384940877947, −5.19119003109069709016037664313, −4.50573274750858261865635783601, −4.14196833507790860893510165511, −3.64336894141285157711146419662, −2.63840021607510864531336217487, −2.14875707598864274370920681680, −1.80651653849649046040952359228, −1.28249196701563611811221350009,
1.28249196701563611811221350009, 1.80651653849649046040952359228, 2.14875707598864274370920681680, 2.63840021607510864531336217487, 3.64336894141285157711146419662, 4.14196833507790860893510165511, 4.50573274750858261865635783601, 5.19119003109069709016037664313, 5.50832957864501331384940877947, 6.08642031414999280199076545080, 6.34102406704256646259256177267, 6.64514109159385122075729483782, 7.47986182454641841972791260896, 7.51418003949278447526066592128, 8.371190256571500973490244011435, 8.594974867359990464099976452449, 8.968122324174631836059259933793, 9.734402582732179603870549351118, 10.09706287191021982127093132817, 10.44075119996207507133698932632
