| L(s) = 1 | − 4-s + 2·5-s − 9-s + 16-s − 2·20-s + 3·25-s + 36-s − 4·41-s − 2·45-s − 49-s − 64-s + 2·80-s + 81-s − 4·89-s − 3·100-s + 4·101-s + 4·109-s − 2·121-s + 4·125-s + ⋯ |
| L(s) = 1 | − 4-s + 2·5-s − 9-s + 16-s − 2·20-s + 3·25-s + 36-s − 4·41-s − 2·45-s − 49-s − 64-s + 2·80-s + 81-s − 4·89-s − 3·100-s + 4·101-s + 4·109-s − 2·121-s + 4·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7329804232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7329804232\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−11.49167277879116772856096559277, −11.27774298483463422992297037087, −10.36318347351254437845862606871, −10.27690357648559879869450191358, −9.849601595597083540294523101414, −9.460954766192921269399508101495, −8.882506539365410725055741343258, −8.463863264496142831702407611184, −8.453314013034598148448683264804, −7.46992445989136278910409620516, −6.83816375826502816996740666978, −6.32033675389585685212976300560, −5.92501627286758805732609489839, −5.35941939797351213158440148615, −5.04149430345979851238375356586, −4.58372416401106402086980854062, −3.43455868965670459222249926831, −3.12699937113720365109990098080, −2.18555724450550750940052168903, −1.47449520356170165005542554715,
1.47449520356170165005542554715, 2.18555724450550750940052168903, 3.12699937113720365109990098080, 3.43455868965670459222249926831, 4.58372416401106402086980854062, 5.04149430345979851238375356586, 5.35941939797351213158440148615, 5.92501627286758805732609489839, 6.32033675389585685212976300560, 6.83816375826502816996740666978, 7.46992445989136278910409620516, 8.453314013034598148448683264804, 8.463863264496142831702407611184, 8.882506539365410725055741343258, 9.460954766192921269399508101495, 9.849601595597083540294523101414, 10.27690357648559879869450191358, 10.36318347351254437845862606871, 11.27774298483463422992297037087, 11.49167277879116772856096559277
