| L(s) = 1 | + 86·9-s + 248·11-s − 6.08e3·19-s + 6.56e3·29-s − 1.14e4·31-s − 1.77e4·41-s + 3.30e4·49-s − 3.37e4·59-s − 3.69e4·61-s − 6.39e4·71-s − 8.91e4·79-s − 5.16e4·81-s − 1.43e5·89-s + 2.13e4·99-s + 2.25e5·109-s − 2.75e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.14e5·169-s − 5.23e5·171-s + ⋯ |
| L(s) = 1 | + 0.353·9-s + 0.617·11-s − 3.86·19-s + 1.44·29-s − 2.14·31-s − 1.65·41-s + 1.96·49-s − 1.26·59-s − 1.27·61-s − 1.50·71-s − 1.60·79-s − 0.874·81-s − 1.92·89-s + 0.218·99-s + 1.81·109-s − 1.71·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.38·169-s − 1.36·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7825353958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7825353958\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 - 86 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 33038 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 124 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 514102 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1404510 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3044 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24270 p^{2} T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3282 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5728 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 32061638 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8886 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 209597542 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 101294882 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 699828390 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 16876 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 18482 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2459007190 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 31960 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4122270190 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 44560 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3340172790 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 71994 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 14786794558 T^{2} + p^{10} T^{4} \) |
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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
−12.03572340098120601598174672203, −11.16993604996919892882464326307, −10.83012660504994142919042566429, −10.33429437644249312575827315683, −10.11132540666063241857693165950, −9.140979277202607640645187476456, −8.761513141881721603336813401091, −8.550444130005606340715493837899, −7.83869722148292122706082441747, −6.91489046709877375782035709337, −6.89938247224806911676175738282, −6.06211529109739476822646204276, −5.73842538736061835062800473690, −4.54273724968943560426844887269, −4.42902416785970411890503377463, −3.74843277682716594817605308180, −2.84593751831303045828783058062, −1.98498027830874329304439730333, −1.53897883558213299010803342532, −0.26108761409163315725902299614,
0.26108761409163315725902299614, 1.53897883558213299010803342532, 1.98498027830874329304439730333, 2.84593751831303045828783058062, 3.74843277682716594817605308180, 4.42902416785970411890503377463, 4.54273724968943560426844887269, 5.73842538736061835062800473690, 6.06211529109739476822646204276, 6.89938247224806911676175738282, 6.91489046709877375782035709337, 7.83869722148292122706082441747, 8.550444130005606340715493837899, 8.761513141881721603336813401091, 9.140979277202607640645187476456, 10.11132540666063241857693165950, 10.33429437644249312575827315683, 10.83012660504994142919042566429, 11.16993604996919892882464326307, 12.03572340098120601598174672203
