| L(s) = 1 | − 2·3-s − 3·4-s − 6·5-s − 3·9-s − 2·11-s + 6·12-s + 12·15-s + 5·16-s + 18·20-s + 4·23-s + 17·25-s + 14·27-s − 20·31-s + 4·33-s + 9·36-s − 4·37-s + 6·44-s + 18·45-s + 14·47-s − 10·48-s − 13·49-s + 16·53-s + 12·55-s − 6·59-s − 36·60-s − 3·64-s + 16·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s − 2.68·5-s − 9-s − 0.603·11-s + 1.73·12-s + 3.09·15-s + 5/4·16-s + 4.02·20-s + 0.834·23-s + 17/5·25-s + 2.69·27-s − 3.59·31-s + 0.696·33-s + 3/2·36-s − 0.657·37-s + 0.904·44-s + 2.68·45-s + 2.04·47-s − 1.44·48-s − 1.85·49-s + 2.19·53-s + 1.61·55-s − 0.781·59-s − 4.64·60-s − 3/8·64-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
| 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
−8.100676573139465817505970839874, −7.32105974546898640977495411592, −7.12147698344619796195085066396, −6.61299535373507474596718204766, −5.62855197375189256680055275378, −5.30227837415656706097307136496, −5.29917071437467895295333587611, −4.57590462337170777040438551304, −3.93822077089834607238244805934, −3.76592143771251579951082054976, −3.30515882404899528069116499186, −2.47732001383873516724842424289, −0.857939810458084365118492515226, 0, 0,
0.857939810458084365118492515226, 2.47732001383873516724842424289, 3.30515882404899528069116499186, 3.76592143771251579951082054976, 3.93822077089834607238244805934, 4.57590462337170777040438551304, 5.29917071437467895295333587611, 5.30227837415656706097307136496, 5.62855197375189256680055275378, 6.61299535373507474596718204766, 7.12147698344619796195085066396, 7.32105974546898640977495411592, 8.100676573139465817505970839874
