| L(s) = 1 | + 3-s + 2·5-s − 5·7-s + 6·11-s + 6·13-s + 2·15-s − 4·17-s + 5·19-s − 5·21-s − 4·23-s + 5·25-s − 27-s + 8·29-s + 7·31-s + 6·33-s − 10·35-s − 9·37-s + 6·39-s − 4·41-s − 2·43-s + 2·47-s + 18·49-s − 4·51-s + 8·53-s + 12·55-s + 5·57-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.88·7-s + 1.80·11-s + 1.66·13-s + 0.516·15-s − 0.970·17-s + 1.14·19-s − 1.09·21-s − 0.834·23-s + 25-s − 0.192·27-s + 1.48·29-s + 1.25·31-s + 1.04·33-s − 1.69·35-s − 1.47·37-s + 0.960·39-s − 0.624·41-s − 0.304·43-s + 0.291·47-s + 18/7·49-s − 0.560·51-s + 1.09·53-s + 1.61·55-s + 0.662·57-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.527317695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.527317695\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−9.661828778204744979263752974985, −9.454157858589829136293294036860, −9.061147635327130878898546492572, −8.618819852881984684475716843559, −8.476886523323609540548834333354, −7.996967516213156089272758234472, −7.05527745628719582604363649878, −6.69569905015701800945830599205, −6.60166636060455075415403691746, −6.35577448308589442350525109577, −5.78321394563122959727903046208, −5.33631258717304183008560037926, −4.71496711532365541581977928842, −3.87557989379165919252853137159, −3.69622942818843229777735936602, −3.43769372299176148214603165299, −2.62488581165896639895207039758, −2.29084217163218853439282551178, −1.31926011431250485134687295627, −0.846566050412001695512001384867,
0.846566050412001695512001384867, 1.31926011431250485134687295627, 2.29084217163218853439282551178, 2.62488581165896639895207039758, 3.43769372299176148214603165299, 3.69622942818843229777735936602, 3.87557989379165919252853137159, 4.71496711532365541581977928842, 5.33631258717304183008560037926, 5.78321394563122959727903046208, 6.35577448308589442350525109577, 6.60166636060455075415403691746, 6.69569905015701800945830599205, 7.05527745628719582604363649878, 7.996967516213156089272758234472, 8.476886523323609540548834333354, 8.618819852881984684475716843559, 9.061147635327130878898546492572, 9.454157858589829136293294036860, 9.661828778204744979263752974985
