| L(s) = 1 | − 2·3-s + 4·5-s − 5·7-s + 3·9-s − 5·11-s − 3·13-s − 8·15-s − 3·17-s − 7·19-s + 10·21-s − 3·23-s + 2·25-s − 4·27-s + 29-s − 5·31-s + 10·33-s − 20·35-s + 5·37-s + 6·39-s − 11·41-s + 8·43-s + 12·45-s + 3·47-s + 7·49-s + 6·51-s − 12·53-s − 20·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 1.88·7-s + 9-s − 1.50·11-s − 0.832·13-s − 2.06·15-s − 0.727·17-s − 1.60·19-s + 2.18·21-s − 0.625·23-s + 2/5·25-s − 0.769·27-s + 0.185·29-s − 0.898·31-s + 1.74·33-s − 3.38·35-s + 0.821·37-s + 0.960·39-s − 1.71·41-s + 1.21·43-s + 1.78·45-s + 0.437·47-s + 49-s + 0.840·51-s − 1.64·53-s − 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 646416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 646416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1790637405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1790637405\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 67 | $C_2$ | \( 1 - 16 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 11 T + 80 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 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
−10.47330179768290229964536643990, −10.07222222908808503930706382522, −9.716077733487740726662263683089, −9.289455297785225072877052958819, −9.277037451098849523596741938201, −8.189151490038912072878799810193, −8.028178539005413540804408531274, −7.14709394731671295485955705971, −6.88616679477676019542069509815, −6.28770276629955051138768216279, −6.15836565015958488463690392088, −5.77039090377263434220253935699, −5.32449758289253266854876728478, −4.77566353187544674384816648908, −4.28799819643648618928183552791, −3.49342363351217763007779990538, −2.78425961494884892274306576719, −2.21218331724074080016703951081, −1.84417672452927925837383579503, −0.20617530664413012890178828043,
0.20617530664413012890178828043, 1.84417672452927925837383579503, 2.21218331724074080016703951081, 2.78425961494884892274306576719, 3.49342363351217763007779990538, 4.28799819643648618928183552791, 4.77566353187544674384816648908, 5.32449758289253266854876728478, 5.77039090377263434220253935699, 6.15836565015958488463690392088, 6.28770276629955051138768216279, 6.88616679477676019542069509815, 7.14709394731671295485955705971, 8.028178539005413540804408531274, 8.189151490038912072878799810193, 9.277037451098849523596741938201, 9.289455297785225072877052958819, 9.716077733487740726662263683089, 10.07222222908808503930706382522, 10.47330179768290229964536643990
