| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 3·9-s − 2·12-s − 14-s + 16-s + 3·18-s − 8·19-s + 2·21-s − 2·24-s − 6·25-s − 4·27-s − 28-s − 4·29-s + 32-s + 3·36-s − 20·37-s − 8·38-s + 2·42-s − 2·48-s + 49-s − 6·50-s + 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 9-s − 0.577·12-s − 0.267·14-s + 1/4·16-s + 0.707·18-s − 1.83·19-s + 0.436·21-s − 0.408·24-s − 6/5·25-s − 0.769·27-s − 0.188·28-s − 0.742·29-s + 0.176·32-s + 1/2·36-s − 3.28·37-s − 1.29·38-s + 0.308·42-s − 0.288·48-s + 1/7·49-s − 0.848·50-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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
−9.415781556927177003100714607194, −8.825241684599749435471219697260, −8.356918966632091922574869118017, −7.67454257840429899018734503053, −6.92538722224681151756066962499, −6.84552480602986516155667793601, −6.18874160468740625251832036222, −5.60435712531698215962545180754, −5.33985014787602837985094112170, −4.60067949543777570406002885677, −3.91638973375108536412463827494, −3.62482887081886485478101246558, −2.39883079209175985717649788244, −1.69840313204406173914058397816, 0,
1.69840313204406173914058397816, 2.39883079209175985717649788244, 3.62482887081886485478101246558, 3.91638973375108536412463827494, 4.60067949543777570406002885677, 5.33985014787602837985094112170, 5.60435712531698215962545180754, 6.18874160468740625251832036222, 6.84552480602986516155667793601, 6.92538722224681151756066962499, 7.67454257840429899018734503053, 8.356918966632091922574869118017, 8.825241684599749435471219697260, 9.415781556927177003100714607194
