| L(s) = 1 | + 5·9-s + 2·11-s − 4·16-s + 14·31-s − 16·41-s + 10·49-s − 10·59-s + 24·61-s − 6·71-s + 20·79-s + 16·81-s − 30·89-s + 10·99-s + 4·101-s − 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 0.603·11-s − 16-s + 2.51·31-s − 2.49·41-s + 10/7·49-s − 1.30·59-s + 3.07·61-s − 0.712·71-s + 2.25·79-s + 16/9·81-s − 3.17·89-s + 1.00·99-s + 0.398·101-s − 1.91·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.702516737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.702516737\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 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
−12.25940612211621853665452299995, −11.53396745702261126366409928745, −11.43558853428207847962679445284, −10.53213938107971233805540401914, −10.06665428588082261076458198889, −10.03750022045939640447522428704, −9.251477131161992413121569130863, −8.889224296787787090698632950554, −8.209335702871744089617690896496, −7.84237457796866250230965741845, −7.01651101672238661047656276596, −6.65108822145973853100510609900, −6.55003528032179780820142489133, −5.45129199703095345621440390589, −4.90979256355504269162896996230, −4.23102062652037313826257902463, −3.96908582963553082050621433685, −2.94748151665832611682961435334, −2.05327083745803521896391993931, −1.14477533681318858572605649219,
1.14477533681318858572605649219, 2.05327083745803521896391993931, 2.94748151665832611682961435334, 3.96908582963553082050621433685, 4.23102062652037313826257902463, 4.90979256355504269162896996230, 5.45129199703095345621440390589, 6.55003528032179780820142489133, 6.65108822145973853100510609900, 7.01651101672238661047656276596, 7.84237457796866250230965741845, 8.209335702871744089617690896496, 8.889224296787787090698632950554, 9.251477131161992413121569130863, 10.03750022045939640447522428704, 10.06665428588082261076458198889, 10.53213938107971233805540401914, 11.43558853428207847962679445284, 11.53396745702261126366409928745, 12.25940612211621853665452299995
