| L(s) = 1 | + 4·5-s − 4·11-s + 12·17-s + 8·19-s − 4·23-s + 4·25-s + 8·37-s + 12·41-s − 8·47-s + 4·53-s − 16·55-s − 8·61-s − 4·71-s − 24·73-s − 16·79-s + 8·83-s + 48·85-s + 20·89-s + 32·95-s − 8·97-s + 4·101-s + 20·107-s + 16·109-s + 20·113-s − 16·115-s − 2·121-s − 12·125-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.20·11-s + 2.91·17-s + 1.83·19-s − 0.834·23-s + 4/5·25-s + 1.31·37-s + 1.87·41-s − 1.16·47-s + 0.549·53-s − 2.15·55-s − 1.02·61-s − 0.474·71-s − 2.80·73-s − 1.80·79-s + 0.878·83-s + 5.20·85-s + 2.11·89-s + 3.28·95-s − 0.812·97-s + 0.398·101-s + 1.93·107-s + 1.53·109-s + 1.88·113-s − 1.49·115-s − 0.181·121-s − 1.07·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.536234404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.536234404\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 4 p T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 24 T + 272 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.938745852718042173722652679845, −7.61521599329506929466557052016, −7.52875208988564582036711018828, −7.38376184462312238094348375669, −6.55664667862839000881003602306, −6.18126367119625366455855257133, −5.81932365823782292937544258576, −5.70013144790019123588953939150, −5.50798605144875358760445915357, −5.14187162300529699490578232013, −4.56361718190662142544979716415, −4.34819313248293558788317763399, −3.44744830979579705900893433965, −3.38799767153823859996051436817, −2.74011801101535437561929336234, −2.71877799592517012978571514122, −1.94028503160594080325178544958, −1.60582103892732957435416612135, −1.09649935208597408983649326140, −0.61504916829402728232577831840,
0.61504916829402728232577831840, 1.09649935208597408983649326140, 1.60582103892732957435416612135, 1.94028503160594080325178544958, 2.71877799592517012978571514122, 2.74011801101535437561929336234, 3.38799767153823859996051436817, 3.44744830979579705900893433965, 4.34819313248293558788317763399, 4.56361718190662142544979716415, 5.14187162300529699490578232013, 5.50798605144875358760445915357, 5.70013144790019123588953939150, 5.81932365823782292937544258576, 6.18126367119625366455855257133, 6.55664667862839000881003602306, 7.38376184462312238094348375669, 7.52875208988564582036711018828, 7.61521599329506929466557052016, 7.938745852718042173722652679845
