| L(s) = 1 | − 4·2-s + 12·4-s − 32·8-s + 18·9-s + 80·16-s − 72·18-s + 48·25-s + 80·29-s − 192·32-s + 216·36-s + 48·37-s − 192·50-s − 180·53-s − 320·58-s + 448·64-s − 576·72-s − 192·74-s + 243·81-s + 576·100-s + 720·106-s − 240·109-s + 60·113-s + 960·116-s + 242·121-s + 127-s − 1.02e3·128-s + 131-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 5·16-s − 4·18-s + 1.91·25-s + 2.75·29-s − 6·32-s + 6·36-s + 1.29·37-s − 3.83·50-s − 3.39·53-s − 5.51·58-s + 7·64-s − 8·72-s − 2.59·74-s + 3·81-s + 5.75·100-s + 6.79·106-s − 2.20·109-s + 0.530·113-s + 8.27·116-s + 2·121-s + 0.00787·127-s − 8·128-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.038998806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038998806\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 240 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 480 T^{2} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1440 T^{2} + p^{4} T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 2640 T^{2} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 10560 T^{2} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12480 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 18720 T^{2} + p^{4} T^{4} \) |
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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
−12.29957483645858595912604475681, −12.05969426271667443711214752790, −11.18474853293253268510591277637, −10.85105768455747840599779210750, −10.42746641950694823089569663269, −9.950743675716325120434786390903, −9.622660598500276716065877357701, −9.136293817203693786761826685093, −8.488329443821822400870134411587, −8.053895047121480890664567163296, −7.54349624694807677832046655208, −7.00447162039958653641435196975, −6.46172645210458395389368768624, −6.31689026847130285081656716190, −5.01109808819709229740684019061, −4.43368957224282451476608778247, −3.25167887480955866489941193263, −2.59062794112887481138031594308, −1.48040477790964852118698579886, −0.923520418246340148554936793549,
0.923520418246340148554936793549, 1.48040477790964852118698579886, 2.59062794112887481138031594308, 3.25167887480955866489941193263, 4.43368957224282451476608778247, 5.01109808819709229740684019061, 6.31689026847130285081656716190, 6.46172645210458395389368768624, 7.00447162039958653641435196975, 7.54349624694807677832046655208, 8.053895047121480890664567163296, 8.488329443821822400870134411587, 9.136293817203693786761826685093, 9.622660598500276716065877357701, 9.950743675716325120434786390903, 10.42746641950694823089569663269, 10.85105768455747840599779210750, 11.18474853293253268510591277637, 12.05969426271667443711214752790, 12.29957483645858595912604475681
