| L(s) = 1 | + 4-s − 2·7-s + 10·13-s + 16-s + 10·19-s + 25-s − 2·28-s + 16·31-s + 4·37-s − 8·43-s − 11·49-s + 10·52-s − 8·61-s + 64-s − 8·67-s + 28·73-s + 10·76-s + 28·79-s − 20·91-s − 32·97-s + 100-s + 34·103-s + 16·109-s − 2·112-s − 22·121-s + 16·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s + 2.77·13-s + 1/4·16-s + 2.29·19-s + 1/5·25-s − 0.377·28-s + 2.87·31-s + 0.657·37-s − 1.21·43-s − 1.57·49-s + 1.38·52-s − 1.02·61-s + 1/8·64-s − 0.977·67-s + 3.27·73-s + 1.14·76-s + 3.15·79-s − 2.09·91-s − 3.24·97-s + 1/10·100-s + 3.35·103-s + 1.53·109-s − 0.188·112-s − 2·121-s + 1.43·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.866295144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.866295144\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−8.312293789871667127278945468914, −7.82291029009474417186931317295, −7.71078313759863912854007134813, −6.67660955331412487149012069261, −6.57624001584826522381764449343, −6.19737888035657731660047131108, −5.85900984964464586455726187470, −4.98203665329191903836820756371, −4.85502898063728844739466830560, −3.67218944522848342314411371887, −3.62440610456498725135061048084, −3.08330236232313765921606709258, −2.47213607274368988105197886301, −1.27415075733749215024234761238, −1.05708300590427691886411977460,
1.05708300590427691886411977460, 1.27415075733749215024234761238, 2.47213607274368988105197886301, 3.08330236232313765921606709258, 3.62440610456498725135061048084, 3.67218944522848342314411371887, 4.85502898063728844739466830560, 4.98203665329191903836820756371, 5.85900984964464586455726187470, 6.19737888035657731660047131108, 6.57624001584826522381764449343, 6.67660955331412487149012069261, 7.71078313759863912854007134813, 7.82291029009474417186931317295, 8.312293789871667127278945468914
