| L(s) = 1 | + 4·5-s + 5·9-s − 6·11-s − 4·16-s + 11·25-s + 10·29-s − 4·31-s − 4·41-s + 20·45-s − 24·55-s + 20·59-s + 16·61-s − 16·71-s + 10·79-s − 16·80-s + 16·81-s − 30·99-s − 24·101-s − 10·109-s + 5·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + 40·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 5/3·9-s − 1.80·11-s − 16-s + 11/5·25-s + 1.85·29-s − 0.718·31-s − 0.624·41-s + 2.98·45-s − 3.23·55-s + 2.60·59-s + 2.04·61-s − 1.89·71-s + 1.12·79-s − 1.78·80-s + 16/9·81-s − 3.01·99-s − 2.38·101-s − 0.957·109-s + 5/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-s + 3.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.968030875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.968030875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−12.87800358646590012494933048348, −11.84499212526274276054351731312, −11.35436837004902089596449340197, −10.48830894694029595254842704192, −10.28435667535972885534450575044, −10.21239243694715159637043713959, −9.552653034685264689496698174365, −9.083276566823832879511463172761, −8.497055539128846024946525050910, −7.919503696406930494753125122485, −7.25555219626301205548661667912, −6.61974670488315296105460364924, −6.57030394030377988870854436187, −5.38402049899582686739742874953, −5.31323077445712023937989250955, −4.64975090428334336306619623857, −3.88899238689639897784968487792, −2.62782325971648773735830990284, −2.34216171330090280540722742525, −1.33621710577323121618905087609,
1.33621710577323121618905087609, 2.34216171330090280540722742525, 2.62782325971648773735830990284, 3.88899238689639897784968487792, 4.64975090428334336306619623857, 5.31323077445712023937989250955, 5.38402049899582686739742874953, 6.57030394030377988870854436187, 6.61974670488315296105460364924, 7.25555219626301205548661667912, 7.919503696406930494753125122485, 8.497055539128846024946525050910, 9.083276566823832879511463172761, 9.552653034685264689496698174365, 10.21239243694715159637043713959, 10.28435667535972885534450575044, 10.48830894694029595254842704192, 11.35436837004902089596449340197, 11.84499212526274276054351731312, 12.87800358646590012494933048348
