| L(s) = 1 | − 8·4-s + 56·7-s + 48·16-s − 476·25-s − 448·28-s + 1.72e3·37-s + 1.42e3·43-s + 1.66e3·49-s − 256·64-s + 2.30e3·67-s − 1.50e3·79-s + 3.80e3·100-s − 560·109-s + 2.68e3·112-s + 3.52e3·121-s + 127-s + 131-s + 137-s + 139-s − 1.37e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.03e3·169-s − 1.13e4·172-s + ⋯ |
| L(s) = 1 | − 4-s + 3.02·7-s + 3/4·16-s − 3.80·25-s − 3.02·28-s + 7.64·37-s + 5.03·43-s + 34/7·49-s − 1/2·64-s + 4.19·67-s − 2.14·79-s + 3.80·100-s − 0.492·109-s + 2.26·112-s + 2.64·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 7.64·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.29·169-s − 5.03·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.851411914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.851411914\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 238 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 1762 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2519 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 1337 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 5606 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 16067 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 36457 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 18515 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 430 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 123970 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 355 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 139154 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 296665 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 70051 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 26038 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 575 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 690541 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 775682 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 376 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 1089706 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 1089151 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1427858 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−7.79080182086822213624884257081, −7.58511523446094236381327063372, −7.55138655246846345912989195451, −7.45658363437171362547282672166, −6.87549106982146400697965500824, −6.22778430692757667382894531091, −6.01093717811126098830667368110, −6.00869440976459068900711869553, −5.78975042995164382576536587417, −5.51697793417976404963740410944, −5.11502799261172477793336496745, −4.82856912989993330335445465439, −4.55088850679565259586614361528, −4.21838560085904082082625349164, −4.13328462992828048623184089797, −3.99467427805227503229224820685, −3.82954598938587230231436918511, −2.89235045243956826248366686383, −2.50402530419515484634089218234, −2.27168681664700489029045656383, −2.23149405589999106718677457500, −1.52768729386082536246122098597, −0.956104823446963746274138862234, −0.943993664872304364908642679048, −0.51670044680637618239447270478,
0.51670044680637618239447270478, 0.943993664872304364908642679048, 0.956104823446963746274138862234, 1.52768729386082536246122098597, 2.23149405589999106718677457500, 2.27168681664700489029045656383, 2.50402530419515484634089218234, 2.89235045243956826248366686383, 3.82954598938587230231436918511, 3.99467427805227503229224820685, 4.13328462992828048623184089797, 4.21838560085904082082625349164, 4.55088850679565259586614361528, 4.82856912989993330335445465439, 5.11502799261172477793336496745, 5.51697793417976404963740410944, 5.78975042995164382576536587417, 6.00869440976459068900711869553, 6.01093717811126098830667368110, 6.22778430692757667382894531091, 6.87549106982146400697965500824, 7.45658363437171362547282672166, 7.55138655246846345912989195451, 7.58511523446094236381327063372, 7.79080182086822213624884257081
