| L(s) = 1 | + 4-s + 9-s − 12·11-s + 4·19-s + 6·29-s − 16·31-s + 36-s + 6·41-s − 12·44-s − 10·49-s + 14·61-s − 64-s − 12·71-s + 4·76-s + 16·79-s + 36·89-s − 12·99-s − 30·101-s − 56·109-s + 6·116-s + 58·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s − 3.61·11-s + 0.917·19-s + 1.11·29-s − 2.87·31-s + 1/6·36-s + 0.937·41-s − 1.80·44-s − 1.42·49-s + 1.79·61-s − 1/8·64-s − 1.42·71-s + 0.458·76-s + 1.80·79-s + 3.81·89-s − 1.20·99-s − 2.98·101-s − 5.36·109-s + 0.557·116-s + 5.27·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01750850070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01750850070\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 14 T^{2} - 1653 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 169 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.55659659326360303034072045564, −6.32502173584417055060241410384, −6.30630340592177263819999125436, −5.65375478358991081435350165736, −5.57888720664673909398696779439, −5.53366151530115960643186963417, −5.23969898136985197182475183324, −5.00976794389812729559730883889, −4.95288303532525647942837085101, −4.91549559466148762156176524879, −4.32541736153463609214917158232, −4.14797206687452650102444477767, −3.73319036892044647069089576359, −3.71564622359059726559688391519, −3.50065826433935028656385380669, −2.90456642338957148776161984674, −2.88468329137275308821534147878, −2.63830693879093670153920312134, −2.53813844881601110681485515356, −2.15674617584918222112873756750, −1.90065909901455752680545245191, −1.51709889824810507236412456790, −1.19939536916582005569639051604, −0.67890398560157800345575475603, −0.02657176528901211753247650572,
0.02657176528901211753247650572, 0.67890398560157800345575475603, 1.19939536916582005569639051604, 1.51709889824810507236412456790, 1.90065909901455752680545245191, 2.15674617584918222112873756750, 2.53813844881601110681485515356, 2.63830693879093670153920312134, 2.88468329137275308821534147878, 2.90456642338957148776161984674, 3.50065826433935028656385380669, 3.71564622359059726559688391519, 3.73319036892044647069089576359, 4.14797206687452650102444477767, 4.32541736153463609214917158232, 4.91549559466148762156176524879, 4.95288303532525647942837085101, 5.00976794389812729559730883889, 5.23969898136985197182475183324, 5.53366151530115960643186963417, 5.57888720664673909398696779439, 5.65375478358991081435350165736, 6.30630340592177263819999125436, 6.32502173584417055060241410384, 6.55659659326360303034072045564
