| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s − 4·13-s − 4·16-s + 2·20-s + 25-s − 8·26-s + 4·31-s − 8·32-s − 12·37-s + 4·41-s − 8·43-s − 2·49-s + 2·50-s − 8·52-s − 12·53-s + 8·62-s − 8·64-s − 4·65-s − 8·67-s + 8·71-s − 24·74-s − 20·79-s − 4·80-s + 8·82-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s − 1.10·13-s − 16-s + 0.447·20-s + 1/5·25-s − 1.56·26-s + 0.718·31-s − 1.41·32-s − 1.97·37-s + 0.624·41-s − 1.21·43-s − 2/7·49-s + 0.282·50-s − 1.10·52-s − 1.64·53-s + 1.01·62-s − 64-s − 0.496·65-s − 0.977·67-s + 0.949·71-s − 2.78·74-s − 2.25·79-s − 0.447·80-s + 0.883·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.146771691994393050815303908361, −7.42695257700080372495319051660, −7.14313300371032718353675494909, −6.64950488221575518059242119575, −6.21010969995511813410702027130, −5.79052586978721763302571297825, −5.16168482517913712450585420243, −4.98234622930131107638460542682, −4.44927217917331309617431380303, −3.94846953218209147042306601798, −3.14742450877212056592783193075, −2.96475114449965439250305675383, −2.17471667752523341729227408401, −1.55986715987930909559071534238, 0,
1.55986715987930909559071534238, 2.17471667752523341729227408401, 2.96475114449965439250305675383, 3.14742450877212056592783193075, 3.94846953218209147042306601798, 4.44927217917331309617431380303, 4.98234622930131107638460542682, 5.16168482517913712450585420243, 5.79052586978721763302571297825, 6.21010969995511813410702027130, 6.64950488221575518059242119575, 7.14313300371032718353675494909, 7.42695257700080372495319051660, 8.146771691994393050815303908361
