| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s + 3·9-s − 4·10-s + 5·16-s + 6·18-s − 6·20-s − 8·23-s − 7·25-s + 8·31-s + 6·32-s + 9·36-s + 6·37-s − 8·40-s − 10·43-s − 6·45-s − 16·46-s − 13·49-s − 14·50-s − 20·59-s − 16·61-s + 16·62-s + 7·64-s + 12·72-s − 20·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s + 9-s − 1.26·10-s + 5/4·16-s + 1.41·18-s − 1.34·20-s − 1.66·23-s − 7/5·25-s + 1.43·31-s + 1.06·32-s + 3/2·36-s + 0.986·37-s − 1.26·40-s − 1.52·43-s − 0.894·45-s − 2.35·46-s − 1.85·49-s − 1.97·50-s − 2.60·59-s − 2.04·61-s + 2.03·62-s + 7/8·64-s + 1.41·72-s − 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1136356 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1136356 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 41 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−7.71485529334377221111399412284, −7.45985560667588965978165906144, −6.80116983878343159848372363275, −6.45868559488600254043933447314, −5.88908450915949561478136129218, −5.83114250852602916241903112470, −4.80012562663540654491464900636, −4.61153453491182141362175201622, −4.21162542836274046123911200331, −3.86264742523296281671271360599, −3.15669098708687979410735548008, −2.87059053260489669092667348906, −1.80351905056271788024052363940, −1.56586460176306848952281318976, 0,
1.56586460176306848952281318976, 1.80351905056271788024052363940, 2.87059053260489669092667348906, 3.15669098708687979410735548008, 3.86264742523296281671271360599, 4.21162542836274046123911200331, 4.61153453491182141362175201622, 4.80012562663540654491464900636, 5.83114250852602916241903112470, 5.88908450915949561478136129218, 6.45868559488600254043933447314, 6.80116983878343159848372363275, 7.45985560667588965978165906144, 7.71485529334377221111399412284
