| L(s) = 1 | − 13·4-s + 50·5-s + 142·9-s + 124·11-s − 87·16-s + 722·19-s − 650·20-s + 1.87e3·25-s − 1.84e3·36-s − 1.61e3·44-s + 7.10e3·45-s + 4.80e3·49-s + 6.20e3·55-s − 1.42e4·61-s + 4.45e3·64-s − 9.38e3·76-s − 4.35e3·80-s + 1.36e4·81-s + 3.61e4·95-s + 1.76e4·99-s − 2.43e4·100-s − 4.01e4·101-s − 1.77e4·121-s + 6.25e4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.812·4-s + 2·5-s + 1.75·9-s + 1.02·11-s − 0.339·16-s + 2·19-s − 1.62·20-s + 3·25-s − 1.42·36-s − 0.832·44-s + 3.50·45-s + 2·49-s + 2.04·55-s − 3.83·61-s + 1.08·64-s − 1.62·76-s − 0.679·80-s + 2.07·81-s + 4·95-s + 1.79·99-s − 2.43·100-s − 3.94·101-s − 1.21·121-s + 4·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.854913189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.854913189\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 13 T^{2} + p^{8} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 142 T^{2} + p^{8} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 62 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 52622 T^{2} + p^{8} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 3237298 T^{2} + p^{8} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 15154382 T^{2} + p^{8} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7138 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 3364622 T^{2} + p^{8} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 60577618 T^{2} + p^{8} T^{4} \) |
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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
−13.63112429628669257412410397800, −13.24889824268321934601030279518, −12.48627851320936524858608739493, −12.24482677212137942277820476746, −11.39281249452580471900645097427, −10.42312666253807203615578862130, −10.31308597026365321269873585868, −9.497509198220176929943932610975, −9.196682763793450800802626843346, −9.149949840979579252312317306240, −7.896901994693737075867863953016, −7.10848190054409813626417636181, −6.71068741944141999428715613552, −5.93142777306281171668841638918, −5.26451363876237482148534361958, −4.64158415850987255923859574541, −3.91912409697826439021313716622, −2.75774852437567556429450832923, −1.55415972787308553030364750114, −1.13405460170998045044931564503,
1.13405460170998045044931564503, 1.55415972787308553030364750114, 2.75774852437567556429450832923, 3.91912409697826439021313716622, 4.64158415850987255923859574541, 5.26451363876237482148534361958, 5.93142777306281171668841638918, 6.71068741944141999428715613552, 7.10848190054409813626417636181, 7.896901994693737075867863953016, 9.149949840979579252312317306240, 9.196682763793450800802626843346, 9.497509198220176929943932610975, 10.31308597026365321269873585868, 10.42312666253807203615578862130, 11.39281249452580471900645097427, 12.24482677212137942277820476746, 12.48627851320936524858608739493, 13.24889824268321934601030279518, 13.63112429628669257412410397800
