| L(s) = 1 | + 4·5-s − 5·7-s − 4·11-s + 3·13-s + 8·17-s + 19-s − 16·23-s + 2·25-s + 4·29-s − 3·31-s − 20·35-s + 37-s + 6·41-s − 11·43-s + 6·47-s + 18·49-s − 12·53-s − 16·55-s + 4·59-s + 6·61-s + 12·65-s − 13·67-s + 20·71-s + 11·73-s + 20·77-s + 3·79-s + 2·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.88·7-s − 1.20·11-s + 0.832·13-s + 1.94·17-s + 0.229·19-s − 3.33·23-s + 2/5·25-s + 0.742·29-s − 0.538·31-s − 3.38·35-s + 0.164·37-s + 0.937·41-s − 1.67·43-s + 0.875·47-s + 18/7·49-s − 1.64·53-s − 2.15·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s − 1.58·67-s + 2.37·71-s + 1.28·73-s + 2.27·77-s + 0.337·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684204505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684204505\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 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
−9.515301336661274854884769863754, −9.000212758456728362621690504352, −8.344517441363426941166211926395, −7.996509242237018635867543329055, −7.84179929845912335778906022086, −7.31060512349369400305750995214, −6.62503749218396645330216282857, −6.30549875346230478326269624513, −6.14493559053094399184237769290, −5.66533765327689565967496171376, −5.53513632282713089785832547663, −5.14810240008522480478616670969, −4.25811942113176240075701198402, −3.71632197440089622758523613608, −3.52510699373237300541551136865, −2.91737663209407363230463544992, −2.39962699984127349239247339258, −2.02380813467065873040384379180, −1.38492303028493436244175965900, −0.43432404837003469094928763357,
0.43432404837003469094928763357, 1.38492303028493436244175965900, 2.02380813467065873040384379180, 2.39962699984127349239247339258, 2.91737663209407363230463544992, 3.52510699373237300541551136865, 3.71632197440089622758523613608, 4.25811942113176240075701198402, 5.14810240008522480478616670969, 5.53513632282713089785832547663, 5.66533765327689565967496171376, 6.14493559053094399184237769290, 6.30549875346230478326269624513, 6.62503749218396645330216282857, 7.31060512349369400305750995214, 7.84179929845912335778906022086, 7.996509242237018635867543329055, 8.344517441363426941166211926395, 9.000212758456728362621690504352, 9.515301336661274854884769863754
