| L(s) = 1 | − 3·3-s + 4-s − 2·5-s + 7-s + 6·9-s − 3·12-s + 6·15-s + 16-s − 6·17-s − 2·20-s − 3·21-s − 7·25-s − 9·27-s + 28-s − 2·35-s + 6·36-s + 6·37-s − 10·43-s − 12·45-s + 26·47-s − 3·48-s − 6·49-s + 18·51-s − 20·59-s + 6·60-s + 6·63-s + 64-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 2·9-s − 0.866·12-s + 1.54·15-s + 1/4·16-s − 1.45·17-s − 0.447·20-s − 0.654·21-s − 7/5·25-s − 1.73·27-s + 0.188·28-s − 0.338·35-s + 36-s + 0.986·37-s − 1.52·43-s − 1.78·45-s + 3.79·47-s − 0.433·48-s − 6/7·49-s + 2.52·51-s − 2.60·59-s + 0.774·60-s + 0.755·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5890106213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5890106213\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.905235966993525833388868947156, −8.225552760361780021957840571261, −7.62508756975103450239324980184, −7.45985560667588965978165906144, −6.92076425331888408807137767822, −6.38286220020503058850747043516, −5.88908450915949561478136129218, −5.73681320363144770957728323698, −4.71843544103535120742978284678, −4.61153453491182141362175201622, −4.05080151028369257637143221969, −3.36991235720395449589306296865, −2.34027315669446164505659313213, −1.64016380827058072854636802791, −0.49679880081187161703829014750,
0.49679880081187161703829014750, 1.64016380827058072854636802791, 2.34027315669446164505659313213, 3.36991235720395449589306296865, 4.05080151028369257637143221969, 4.61153453491182141362175201622, 4.71843544103535120742978284678, 5.73681320363144770957728323698, 5.88908450915949561478136129218, 6.38286220020503058850747043516, 6.92076425331888408807137767822, 7.45985560667588965978165906144, 7.62508756975103450239324980184, 8.225552760361780021957840571261, 8.905235966993525833388868947156
