| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s + 2·5-s − 4·6-s + 3·9-s + 4·10-s − 4·12-s − 4·15-s − 4·16-s + 6·17-s + 6·18-s + 19-s + 4·20-s − 7·25-s − 4·27-s − 8·30-s − 4·31-s − 8·32-s + 12·34-s + 6·36-s + 2·38-s + 6·45-s + 8·48-s − 5·49-s − 14·50-s − 12·51-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s − 1.63·6-s + 9-s + 1.26·10-s − 1.15·12-s − 1.03·15-s − 16-s + 1.45·17-s + 1.41·18-s + 0.229·19-s + 0.894·20-s − 7/5·25-s − 0.769·27-s − 1.46·30-s − 0.718·31-s − 1.41·32-s + 2.05·34-s + 36-s + 0.324·38-s + 0.894·45-s + 1.15·48-s − 5/7·49-s − 1.97·50-s − 1.68·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−7.75525037135579232003734806978, −7.06801910593054313669065331224, −7.04173846168624158743830695656, −6.11989947342759976206118717321, −6.02567308265222601671827257647, −5.52130085797761183807247002982, −5.49960710074578838011452836914, −4.80392893944635697455333037352, −4.29287803140015808576580756413, −3.90431057417865746907593992810, −3.22481329079945918066760847835, −2.76056898471702515117641585129, −1.85542976329431191785537331479, −1.37927155401612470737240964655, 0,
1.37927155401612470737240964655, 1.85542976329431191785537331479, 2.76056898471702515117641585129, 3.22481329079945918066760847835, 3.90431057417865746907593992810, 4.29287803140015808576580756413, 4.80392893944635697455333037352, 5.49960710074578838011452836914, 5.52130085797761183807247002982, 6.02567308265222601671827257647, 6.11989947342759976206118717321, 7.04173846168624158743830695656, 7.06801910593054313669065331224, 7.75525037135579232003734806978
