| L(s) = 1 | − 729·9-s − 2.57e4·19-s + 4.78e4·31-s − 2.31e5·49-s + 1.25e5·61-s + 4.09e5·79-s + 5.31e5·81-s + 2.69e5·109-s + 3.54e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.78e6·169-s + 1.87e7·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s − 3.74·19-s + 1.60·31-s − 1.96·49-s + 0.555·61-s + 0.830·79-s + 81-s + 0.207·109-s + 2·121-s − 0.577·169-s + 3.74·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1420394008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1420394008\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 231191 T^{2} + p^{12} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2786111 T^{2} + p^{12} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 12851 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 23939 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2826257618 T^{2} + p^{12} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 11064958631 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 62999 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 10741538809 T^{2} + p^{12} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 104459767778 T^{2} + p^{12} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 204622 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 742191991631 T^{2} + p^{12} T^{4} \) |
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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
−10.93343574152071434970363372896, −10.52714724324689030750685003879, −10.10391232948575430714883209386, −9.555847590083928712573717810074, −8.778765443405965148634537677330, −8.677514996895365815251523843972, −8.115143053298663890277038169065, −7.85940395030995570290654493902, −6.79367082601275805879808913186, −6.53154537700095469941176329759, −6.15147686161836196435312550924, −5.59294204323750410486222883398, −4.65666794448894851645510735165, −4.55461782256696118084133033279, −3.75608479147774901939188350050, −3.10264390655278473702034651372, −2.29484190504114465886213311098, −2.09396543205629066527088886469, −1.04730633179360075767908186296, −0.097343642490893291505318998229,
0.097343642490893291505318998229, 1.04730633179360075767908186296, 2.09396543205629066527088886469, 2.29484190504114465886213311098, 3.10264390655278473702034651372, 3.75608479147774901939188350050, 4.55461782256696118084133033279, 4.65666794448894851645510735165, 5.59294204323750410486222883398, 6.15147686161836196435312550924, 6.53154537700095469941176329759, 6.79367082601275805879808913186, 7.85940395030995570290654493902, 8.115143053298663890277038169065, 8.677514996895365815251523843972, 8.778765443405965148634537677330, 9.555847590083928712573717810074, 10.10391232948575430714883209386, 10.52714724324689030750685003879, 10.93343574152071434970363372896
