| L(s) = 1 | − 8·7-s − 14·25-s + 28·37-s + 8·43-s + 34·49-s − 8·67-s + 40·79-s + 52·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 112·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 2.79·25-s + 4.60·37-s + 1.21·43-s + 34/7·49-s − 0.977·67-s + 4.50·79-s + 4.98·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 8.46·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330143512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330143512\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
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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
−5.99199822920316607045620948768, −5.97999394463375606826302926304, −5.90286518911750377260681662107, −5.82034701978059741727132604678, −5.76600076727854528197094724023, −5.02061166237691540840470232124, −4.97298416771052609221957228324, −4.69268245711063308092133663035, −4.51353866304988817401188276589, −4.12612018602031606189479632411, −4.09788127557668054129975893460, −4.01514851054813665090003390018, −3.51703100978515882937397451218, −3.34179720937903262778843681848, −3.23407553226875657623405817705, −3.09301302687798773058509012169, −2.87887968375824336203322480100, −2.32852795461321131309851571246, −2.15135714466086062282842259098, −2.12393609344801081837188163161, −1.98201862986420093383132169337, −1.13870564884757769888179378592, −0.78166000136720687015148149888, −0.64600425920769073754122259475, −0.36103029811922878218604411531,
0.36103029811922878218604411531, 0.64600425920769073754122259475, 0.78166000136720687015148149888, 1.13870564884757769888179378592, 1.98201862986420093383132169337, 2.12393609344801081837188163161, 2.15135714466086062282842259098, 2.32852795461321131309851571246, 2.87887968375824336203322480100, 3.09301302687798773058509012169, 3.23407553226875657623405817705, 3.34179720937903262778843681848, 3.51703100978515882937397451218, 4.01514851054813665090003390018, 4.09788127557668054129975893460, 4.12612018602031606189479632411, 4.51353866304988817401188276589, 4.69268245711063308092133663035, 4.97298416771052609221957228324, 5.02061166237691540840470232124, 5.76600076727854528197094724023, 5.82034701978059741727132604678, 5.90286518911750377260681662107, 5.97999394463375606826302926304, 5.99199822920316607045620948768
