| L(s) = 1 | − 4·5-s + 2·9-s + 10·25-s + 8·29-s − 8·45-s + 20·49-s + 40·53-s + 40·73-s − 5·81-s − 56·97-s + 72·101-s + 36·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2/3·9-s + 2·25-s + 1.48·29-s − 1.19·45-s + 20/7·49-s + 5.49·53-s + 4.68·73-s − 5/9·81-s − 5.68·97-s + 7.16·101-s + 3.27·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.030257180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.030257180\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
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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
−7.29584987942659229016975095373, −6.97651921962516698362626666600, −6.79581718641471765254153249095, −6.67673581016367996044427030021, −6.48390110024736871843362987081, −5.88123190369505150680465961096, −5.86702529935699244285319371130, −5.52085984458008069685902947328, −5.45440562769254213522818430190, −4.91146767554922102664139625577, −4.87969923338213709674395187584, −4.62127726278236608139524778222, −4.18804273383760282128669719657, −4.09208062943069456411733875549, −3.85774180610164283782017407454, −3.75427453299980610537228553763, −3.41293620261775500232256219069, −3.05328334719730069449245723287, −2.71922365485351590949553886827, −2.35750430592623377881582532157, −2.24032569123476888842818649292, −1.81745722432679392517838421018, −1.00505730797205185834451090327, −0.76152721493161220013518120062, −0.67142849882816230004825533660,
0.67142849882816230004825533660, 0.76152721493161220013518120062, 1.00505730797205185834451090327, 1.81745722432679392517838421018, 2.24032569123476888842818649292, 2.35750430592623377881582532157, 2.71922365485351590949553886827, 3.05328334719730069449245723287, 3.41293620261775500232256219069, 3.75427453299980610537228553763, 3.85774180610164283782017407454, 4.09208062943069456411733875549, 4.18804273383760282128669719657, 4.62127726278236608139524778222, 4.87969923338213709674395187584, 4.91146767554922102664139625577, 5.45440562769254213522818430190, 5.52085984458008069685902947328, 5.86702529935699244285319371130, 5.88123190369505150680465961096, 6.48390110024736871843362987081, 6.67673581016367996044427030021, 6.79581718641471765254153249095, 6.97651921962516698362626666600, 7.29584987942659229016975095373
