| L(s) = 1 | − 3-s + 5-s + 3·9-s − 3·11-s − 2·13-s − 15-s − 5·17-s − 6·19-s − 8·27-s − 10·29-s + 2·31-s + 3·33-s + 4·37-s + 2·39-s + 4·41-s + 20·43-s + 3·45-s − 9·47-s + 5·51-s − 6·53-s − 3·55-s + 6·57-s − 6·59-s − 12·61-s − 2·65-s + 2·67-s − 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 9-s − 0.904·11-s − 0.554·13-s − 0.258·15-s − 1.21·17-s − 1.37·19-s − 1.53·27-s − 1.85·29-s + 0.359·31-s + 0.522·33-s + 0.657·37-s + 0.320·39-s + 0.624·41-s + 3.04·43-s + 0.447·45-s − 1.31·47-s + 0.700·51-s − 0.824·53-s − 0.404·55-s + 0.794·57-s − 0.781·59-s − 1.53·61-s − 0.248·65-s + 0.244·67-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4417239950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4417239950\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.690182639471531676258020929056, −8.947384973896726967330054325812, −8.873082516822341898584740381467, −7.930501597722165240215229983068, −7.86291934283764993665309411163, −7.27601290637810060246760708892, −7.24602638786825232450904286351, −6.42461629134750665337277215792, −6.23309982014631108005216564193, −5.79183034374521121195258338485, −5.52139184562914501173299934685, −4.73695650391501476229455722819, −4.63743975748303804781087023258, −4.07550742387147876723865384463, −3.81345955315236696091585750899, −2.66349300070233139355651402306, −2.65422559635018364511904999340, −1.85752227474570253921362014755, −1.48148490884598625019512071097, −0.23881461361178018659664430315,
0.23881461361178018659664430315, 1.48148490884598625019512071097, 1.85752227474570253921362014755, 2.65422559635018364511904999340, 2.66349300070233139355651402306, 3.81345955315236696091585750899, 4.07550742387147876723865384463, 4.63743975748303804781087023258, 4.73695650391501476229455722819, 5.52139184562914501173299934685, 5.79183034374521121195258338485, 6.23309982014631108005216564193, 6.42461629134750665337277215792, 7.24602638786825232450904286351, 7.27601290637810060246760708892, 7.86291934283764993665309411163, 7.930501597722165240215229983068, 8.873082516822341898584740381467, 8.947384973896726967330054325812, 9.690182639471531676258020929056
