| L(s) = 1 | + 53·9-s + 90·11-s + 182·19-s − 288·29-s + 52·31-s − 918·41-s + 10·49-s + 144·59-s − 236·61-s + 216·71-s + 1.79e3·79-s + 2.08e3·81-s − 702·89-s + 4.77e3·99-s − 1.90e3·101-s + 1.60e3·109-s + 3.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.45e3·169-s + ⋯ |
| L(s) = 1 | + 1.96·9-s + 2.46·11-s + 2.19·19-s − 1.84·29-s + 0.301·31-s − 3.49·41-s + 0.0291·49-s + 0.317·59-s − 0.495·61-s + 0.361·71-s + 2.55·79-s + 2.85·81-s − 0.836·89-s + 4.84·99-s − 1.87·101-s + 1.40·109-s + 2.56·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.11·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.906233202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.906233202\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 53 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 45 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2458 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3863 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 144 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 459 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 52586 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 11378 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 13610 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 118 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 538525 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 688633 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 898 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 284245 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 351 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1676350 T^{2} + p^{6} T^{4} \) |
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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
−13.72918646526290778297024060138, −13.23685780285049394962348753335, −12.42442482096761679933996178079, −12.12961248404188290857981544684, −11.57849437813299749432542718192, −11.21835019734493481088449009373, −10.13399282547714886175612445311, −9.949827041776075875339728072208, −9.233911830151405501417135977977, −9.099306185678696802390083690620, −7.985330032480269273994850694375, −7.38174822065435501609993085107, −6.79863395467540329750603402373, −6.53777620311412573802084483033, −5.41007124147762024486604492193, −4.74320459809182051351148795248, −3.70804106047148059952631965846, −3.63260068562254612262995119738, −1.69036601213490694776935665105, −1.20542120539560899129498037758,
1.20542120539560899129498037758, 1.69036601213490694776935665105, 3.63260068562254612262995119738, 3.70804106047148059952631965846, 4.74320459809182051351148795248, 5.41007124147762024486604492193, 6.53777620311412573802084483033, 6.79863395467540329750603402373, 7.38174822065435501609993085107, 7.985330032480269273994850694375, 9.099306185678696802390083690620, 9.233911830151405501417135977977, 9.949827041776075875339728072208, 10.13399282547714886175612445311, 11.21835019734493481088449009373, 11.57849437813299749432542718192, 12.12961248404188290857981544684, 12.42442482096761679933996178079, 13.23685780285049394962348753335, 13.72918646526290778297024060138
