| L(s) = 1 | − 5-s + 9-s + 25-s − 10·29-s − 4·41-s − 45-s − 10·49-s − 4·61-s − 30·73-s + 81-s − 10·97-s + 14·101-s + 20·113-s + 2·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 10·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1/3·9-s + 1/5·25-s − 1.85·29-s − 0.624·41-s − 0.149·45-s − 1.42·49-s − 0.512·61-s − 3.51·73-s + 1/9·81-s − 1.01·97-s + 1.39·101-s + 1.88·113-s + 2/11·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.830·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.705751130637767532881753293364, −8.149753965469963649287254630443, −7.54760960341807819712643010591, −7.35228471521914928927322690756, −6.85209786824480812231825511098, −6.16083103132214126740432572169, −5.82237125931266815844841409905, −5.16610287757985331289999376203, −4.62328348482859635122422749526, −4.15169852040612839077521408925, −3.47760041033211219936487607505, −3.05174692607622226606918418410, −2.08765291695643069537999385769, −1.40456156081540345745297569971, 0,
1.40456156081540345745297569971, 2.08765291695643069537999385769, 3.05174692607622226606918418410, 3.47760041033211219936487607505, 4.15169852040612839077521408925, 4.62328348482859635122422749526, 5.16610287757985331289999376203, 5.82237125931266815844841409905, 6.16083103132214126740432572169, 6.85209786824480812231825511098, 7.35228471521914928927322690756, 7.54760960341807819712643010591, 8.149753965469963649287254630443, 8.705751130637767532881753293364
