| L(s) = 1 | − 72·11-s + 200·19-s − 468·29-s − 32·31-s − 180·41-s + 622·49-s − 1.36e3·59-s + 844·61-s + 720·71-s − 1.02e3·79-s − 1.26e3·89-s − 1.11e3·101-s − 3.24e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.29e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.97·11-s + 2.41·19-s − 2.99·29-s − 0.185·31-s − 0.685·41-s + 1.81·49-s − 3.01·59-s + 1.77·61-s + 1.20·71-s − 1.45·79-s − 1.50·89-s − 1.09·101-s − 2.85·109-s + 0.921·121-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 + 1.95·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5960897684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5960897684\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4294 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9502 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 50230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 45290 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 21022 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 684 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 422 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 491302 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 777358 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 512 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 267770 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 714430 T^{2} + p^{6} T^{4} \) |
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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
−9.987229847604555702037506687451, −9.415862468905478396139051004638, −9.289807869095835878138461353066, −8.712223993140354364054080434338, −8.099282683035357314673498208106, −7.62361154736511932299110106330, −7.61779307590673668924201597645, −7.11960714184046959892873589543, −6.60130456414650682455309609079, −5.65129207613844891936159037045, −5.56943809869186732648341564406, −5.32965574008324861582784947859, −4.76575233532445731382951806425, −3.95981512948388568702326972753, −3.59557748071621097046319719558, −2.90509752027566809105551137149, −2.59837407138362409218810670935, −1.80137887952025724190745892935, −1.18012993954132801611757528065, −0.20640262241123807863596677616,
0.20640262241123807863596677616, 1.18012993954132801611757528065, 1.80137887952025724190745892935, 2.59837407138362409218810670935, 2.90509752027566809105551137149, 3.59557748071621097046319719558, 3.95981512948388568702326972753, 4.76575233532445731382951806425, 5.32965574008324861582784947859, 5.56943809869186732648341564406, 5.65129207613844891936159037045, 6.60130456414650682455309609079, 7.11960714184046959892873589543, 7.61779307590673668924201597645, 7.62361154736511932299110106330, 8.099282683035357314673498208106, 8.712223993140354364054080434338, 9.289807869095835878138461353066, 9.415862468905478396139051004638, 9.987229847604555702037506687451
