| L(s) = 1 | − 6·9-s − 192·29-s + 132·41-s − 76·49-s − 152·61-s − 135·81-s − 348·89-s − 168·101-s + 232·109-s − 266·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 124·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 6.62·29-s + 3.21·41-s − 1.55·49-s − 2.49·61-s − 5/3·81-s − 3.91·89-s − 1.66·101-s + 2.12·109-s − 2.19·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.733·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2631430036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2631430036\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 133 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 353 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 347 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 998 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 422 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2638 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 33 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4178 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4718 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 5603 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 4082 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10633 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 10982 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13643 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 87 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 6718 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.39455514362583267214071117798, −6.18977039868320919050752509999, −5.94363066743398669093088876274, −5.83492115815163910189740505673, −5.75102941605345725512902450099, −5.51734551875569669528257526177, −5.29427686925085155228779869323, −5.05095204708422267275430325455, −4.71596805036242745330958941323, −4.59869851624030816619910812754, −4.09123257370951065900690469635, −3.96892353268548356174172564752, −3.88689899157935449107905007840, −3.68630032555508825848437707682, −3.40126545421510585696796458976, −2.89605907136167870630389803603, −2.76255100911814259546111199604, −2.70643464575483640511428446254, −2.17432202706091098180604775635, −1.90890127065785973603918165216, −1.65057495812841922936683716849, −1.39979129139849470909811851194, −1.13465012662652418593623766541, −0.31689888532818608774303986169, −0.12247708277165104184218534983,
0.12247708277165104184218534983, 0.31689888532818608774303986169, 1.13465012662652418593623766541, 1.39979129139849470909811851194, 1.65057495812841922936683716849, 1.90890127065785973603918165216, 2.17432202706091098180604775635, 2.70643464575483640511428446254, 2.76255100911814259546111199604, 2.89605907136167870630389803603, 3.40126545421510585696796458976, 3.68630032555508825848437707682, 3.88689899157935449107905007840, 3.96892353268548356174172564752, 4.09123257370951065900690469635, 4.59869851624030816619910812754, 4.71596805036242745330958941323, 5.05095204708422267275430325455, 5.29427686925085155228779869323, 5.51734551875569669528257526177, 5.75102941605345725512902450099, 5.83492115815163910189740505673, 5.94363066743398669093088876274, 6.18977039868320919050752509999, 6.39455514362583267214071117798
