| L(s) = 1 | − 6·9-s − 8·13-s + 24·37-s + 20·41-s + 14·49-s + 8·53-s + 27·81-s + 20·89-s + 48·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2·9-s − 2.21·13-s + 3.94·37-s + 3.12·41-s + 2·49-s + 1.09·53-s + 3·81-s + 2.11·89-s + 4.43·117-s + 2·121-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 + 1.69·169-s + 0.0760·173-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944596205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944596205\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.864229234684621713217378287232, −8.566839174917558726316243266225, −7.896149526479872609028048004255, −7.80852877735676114772836925711, −7.39303259468716917458983305383, −7.25438679845608531451070043837, −6.38193377788816080786625430517, −6.21341210768247059980174048170, −5.70510013203006760095319576936, −5.65653825307123333157078792124, −4.92256337003355570242357659489, −4.76958668077760820362136380922, −4.06320426163702879380594545379, −3.97484686586189709183783656791, −2.89680729236596222818665924399, −2.84565555392979675698912581105, −2.35382097286369136464435644832, −2.20606770134580424980998439924, −0.76679370213445048722903756590, −0.61710629357465666210310724734,
0.61710629357465666210310724734, 0.76679370213445048722903756590, 2.20606770134580424980998439924, 2.35382097286369136464435644832, 2.84565555392979675698912581105, 2.89680729236596222818665924399, 3.97484686586189709183783656791, 4.06320426163702879380594545379, 4.76958668077760820362136380922, 4.92256337003355570242357659489, 5.65653825307123333157078792124, 5.70510013203006760095319576936, 6.21341210768247059980174048170, 6.38193377788816080786625430517, 7.25438679845608531451070043837, 7.39303259468716917458983305383, 7.80852877735676114772836925711, 7.896149526479872609028048004255, 8.566839174917558726316243266225, 8.864229234684621713217378287232
