| L(s) = 1 | + 4·3-s − 4·5-s + 6·9-s − 16·15-s + 11·25-s − 4·27-s + 8·31-s − 12·37-s − 4·41-s + 16·43-s − 24·45-s + 49-s − 20·53-s − 24·67-s + 44·75-s − 16·79-s − 37·81-s + 12·83-s + 20·89-s + 32·93-s − 24·107-s − 48·111-s − 22·121-s − 16·123-s − 24·125-s + 127-s + 64·129-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 2·9-s − 4.13·15-s + 11/5·25-s − 0.769·27-s + 1.43·31-s − 1.97·37-s − 0.624·41-s + 2.43·43-s − 3.57·45-s + 1/7·49-s − 2.74·53-s − 2.93·67-s + 5.08·75-s − 1.80·79-s − 4.11·81-s + 1.31·83-s + 2.11·89-s + 3.31·93-s − 2.32·107-s − 4.55·111-s − 2·121-s − 1.44·123-s − 2.14·125-s + 0.0887·127-s + 5.63·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−8.188057214509100026281306554311, −7.79778103593305486363806212850, −7.58850489526077960653308183172, −7.14438434722230698367292292955, −6.55227860254673854328689078750, −5.95605621693662308440883825196, −5.22036655714483924618943204859, −4.50185552371175935345657331179, −4.23519781761454872386621187276, −3.62963911580372546953600653852, −3.14137792992390816749783057891, −2.98337737281168113218499349831, −2.27755449098586308117555211465, −1.41272608691567798293346783630, 0,
1.41272608691567798293346783630, 2.27755449098586308117555211465, 2.98337737281168113218499349831, 3.14137792992390816749783057891, 3.62963911580372546953600653852, 4.23519781761454872386621187276, 4.50185552371175935345657331179, 5.22036655714483924618943204859, 5.95605621693662308440883825196, 6.55227860254673854328689078750, 7.14438434722230698367292292955, 7.58850489526077960653308183172, 7.79778103593305486363806212850, 8.188057214509100026281306554311
