| L(s) = 1 | + 2·2-s + 3·4-s − 3·5-s + 4·8-s − 3·9-s − 6·10-s + 3·11-s + 5·13-s + 5·16-s + 3·17-s − 6·18-s + 5·19-s − 9·20-s + 6·22-s + 3·23-s + 5·25-s + 10·26-s + 3·29-s + 8·31-s + 6·32-s + 6·34-s − 9·36-s + 7·37-s + 10·38-s − 12·40-s − 9·41-s − 11·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.34·5-s + 1.41·8-s − 9-s − 1.89·10-s + 0.904·11-s + 1.38·13-s + 5/4·16-s + 0.727·17-s − 1.41·18-s + 1.14·19-s − 2.01·20-s + 1.27·22-s + 0.625·23-s + 25-s + 1.96·26-s + 0.557·29-s + 1.43·31-s + 1.06·32-s + 1.02·34-s − 3/2·36-s + 1.15·37-s + 1.62·38-s − 1.89·40-s − 1.40·41-s − 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.613492971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.613492971\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−10.45894240057243215525564160585, −10.21247363763923301307201137892, −9.378136630421024885442993889799, −9.088659824651638186147102741117, −8.500307128074790691881502643838, −8.126784336166302348008962832565, −7.73781825282072337968316940521, −7.44056363180921246131360944311, −6.63151702010508978452651763437, −6.27251888914789782586974806250, −6.24461950523342558662833986159, −5.39726015866936622502436385649, −4.81978626929117199288236542778, −4.75140182761791417006654958169, −3.74504527436505319337060098467, −3.68633543393850724388050810456, −3.10818549786906463443512333836, −2.80975501401100818542403723629, −1.57491184147247087469499887191, −0.907574533911859653391352160720,
0.907574533911859653391352160720, 1.57491184147247087469499887191, 2.80975501401100818542403723629, 3.10818549786906463443512333836, 3.68633543393850724388050810456, 3.74504527436505319337060098467, 4.75140182761791417006654958169, 4.81978626929117199288236542778, 5.39726015866936622502436385649, 6.24461950523342558662833986159, 6.27251888914789782586974806250, 6.63151702010508978452651763437, 7.44056363180921246131360944311, 7.73781825282072337968316940521, 8.126784336166302348008962832565, 8.500307128074790691881502643838, 9.088659824651638186147102741117, 9.378136630421024885442993889799, 10.21247363763923301307201137892, 10.45894240057243215525564160585
