| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 6·13-s + 16-s − 8·17-s − 12·23-s + 4·27-s + 8·29-s − 3·36-s + 12·39-s + 24·43-s + 2·48-s + 14·49-s − 16·51-s − 6·52-s + 4·53-s − 20·61-s − 64-s + 8·68-s − 24·69-s − 16·79-s + 5·81-s + 16·87-s + 12·92-s + 16·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1.66·13-s + 1/4·16-s − 1.94·17-s − 2.50·23-s + 0.769·27-s + 1.48·29-s − 1/2·36-s + 1.92·39-s + 3.65·43-s + 0.288·48-s + 2·49-s − 2.24·51-s − 0.832·52-s + 0.549·53-s − 2.56·61-s − 1/8·64-s + 0.970·68-s − 2.88·69-s − 1.80·79-s + 5/9·81-s + 1.71·87-s + 1.25·92-s + 1.59·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.118067188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.118067188\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.187613576960461566352025134747, −9.019722307439620712715969406652, −8.608139741057166439950524631461, −8.363539602907481682457783862403, −7.891830857093789402327855086608, −7.59378688054193530172664770070, −7.17256180147514723477012595084, −6.41601366300404473739365371090, −6.39445224184079194766537324328, −5.78747342078323616494685219749, −5.55735063006930008333610860599, −4.48101267794303323257163145678, −4.42704391260931184504391954888, −3.90779067260964431617456351828, −3.87800635582167315382578870137, −2.84803387063646909688736294825, −2.64932643699283655235127342336, −2.02271107803511629709079751488, −1.43611284685129411138177216473, −0.61343835805002563251832578519,
0.61343835805002563251832578519, 1.43611284685129411138177216473, 2.02271107803511629709079751488, 2.64932643699283655235127342336, 2.84803387063646909688736294825, 3.87800635582167315382578870137, 3.90779067260964431617456351828, 4.42704391260931184504391954888, 4.48101267794303323257163145678, 5.55735063006930008333610860599, 5.78747342078323616494685219749, 6.39445224184079194766537324328, 6.41601366300404473739365371090, 7.17256180147514723477012595084, 7.59378688054193530172664770070, 7.891830857093789402327855086608, 8.363539602907481682457783862403, 8.608139741057166439950524631461, 9.019722307439620712715969406652, 9.187613576960461566352025134747
