| L(s) = 1 | + 4-s − 2·7-s − 6·13-s − 3·16-s − 19-s − 4·25-s − 2·28-s − 6·31-s − 4·37-s − 4·43-s − 3·49-s − 6·52-s + 12·61-s − 7·64-s − 2·67-s − 76-s + 4·79-s + 12·91-s + 18·97-s − 4·100-s − 24·103-s + 14·109-s + 6·112-s − 14·121-s − 6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 1.66·13-s − 3/4·16-s − 0.229·19-s − 4/5·25-s − 0.377·28-s − 1.07·31-s − 0.657·37-s − 0.609·43-s − 3/7·49-s − 0.832·52-s + 1.53·61-s − 7/8·64-s − 0.244·67-s − 0.114·76-s + 0.450·79-s + 1.25·91-s + 1.82·97-s − 2/5·100-s − 2.36·103-s + 1.34·109-s + 0.566·112-s − 1.27·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.451808115510595304934440107969, −9.284493818243571866566646306503, −8.523866780545975415324122899128, −7.962936558854149882361927378320, −7.33995880204398288349734876353, −7.00108325834014391470111729249, −6.56218917408877805692663255116, −5.91990419890763971288746040554, −5.26463306290841073268659490826, −4.75590964208523306519663302548, −3.97450961899134079626108319531, −3.28792973835471695930734978908, −2.48940279941837917395101269178, −1.91806497648452369136382678113, 0,
1.91806497648452369136382678113, 2.48940279941837917395101269178, 3.28792973835471695930734978908, 3.97450961899134079626108319531, 4.75590964208523306519663302548, 5.26463306290841073268659490826, 5.91990419890763971288746040554, 6.56218917408877805692663255116, 7.00108325834014391470111729249, 7.33995880204398288349734876353, 7.962936558854149882361927378320, 8.523866780545975415324122899128, 9.284493818243571866566646306503, 9.451808115510595304934440107969
