| L(s) = 1 | − 12·2-s + 54·3-s − 20·4-s + 250·5-s − 648·6-s + 686·7-s + 672·8-s + 2.18e3·9-s − 3.00e3·10-s − 1.09e4·11-s − 1.08e3·12-s − 2.79e3·13-s − 8.23e3·14-s + 1.35e4·15-s − 2.73e3·16-s − 8.28e3·17-s − 2.62e4·18-s − 8.09e3·19-s − 5.00e3·20-s + 3.70e4·21-s + 1.31e5·22-s − 9.09e4·23-s + 3.62e4·24-s + 4.68e4·25-s + 3.35e4·26-s + 7.87e4·27-s − 1.37e4·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1.15·3-s − 0.156·4-s + 0.894·5-s − 1.22·6-s + 0.755·7-s + 0.464·8-s + 9-s − 0.948·10-s − 2.48·11-s − 0.180·12-s − 0.352·13-s − 0.801·14-s + 1.03·15-s − 0.166·16-s − 0.408·17-s − 1.06·18-s − 0.270·19-s − 0.139·20-s + 0.872·21-s + 2.63·22-s − 1.55·23-s + 0.535·24-s + 3/5·25-s + 0.374·26-s + 0.769·27-s − 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 3 p^{2} T + 41 p^{2} T^{2} + 3 p^{9} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 10976 T + 68803686 T^{2} + 10976 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2796 T + 126859566 T^{2} + 2796 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8284 T + 47624614 p T^{2} + 8284 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8096 T + 1803475414 T^{2} + 8096 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 90976 T + 7977553070 T^{2} + 90976 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12532 T + 31965988526 T^{2} + 12532 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 117960 T + 9090248910 T^{2} + 117960 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 174212 T + 106259379454 T^{2} + 174212 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 492700 T + 358124883062 T^{2} + 492700 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 661176 T + 502549980726 T^{2} + 661176 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1675408 T + 1714888352190 T^{2} + 1675408 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 555436 T - 793299419970 T^{2} + 555436 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3121176 T + 6648375612854 T^{2} + 3121176 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 511588 T + 1747585779726 T^{2} + 511588 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 252728 T + 12123587827590 T^{2} + 252728 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1099336 T + 18156240823806 T^{2} + 1099336 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5012588 T + 28363475914198 T^{2} - 5012588 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 83648 T + 38379183136222 T^{2} - 83648 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4404184 T + 30645544257030 T^{2} + 4404184 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4381564 T + 93146164629590 T^{2} + 4381564 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8539828 T + 4428577747174 T^{2} + 8539828 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.23107327955428078505588443207, −11.41012945784341050027251651536, −10.62583505301402662394441528983, −10.34688336473285712867502854354, −9.701101184568594692011608215984, −9.621349777941533985981367704140, −8.595108553937894798103835727892, −8.465802402483926236510824704457, −7.916065177931350363735154918886, −7.56292576532902279117260788537, −6.64404285394586224726158658695, −5.79593439519103874850396460804, −4.86391845645128256096464571519, −4.79404598866282585148191269664, −3.45991199962996539324213027092, −2.70658491673205986314339038043, −2.01088633621629174042378525279, −1.58871829284713238397027166937, 0, 0,
1.58871829284713238397027166937, 2.01088633621629174042378525279, 2.70658491673205986314339038043, 3.45991199962996539324213027092, 4.79404598866282585148191269664, 4.86391845645128256096464571519, 5.79593439519103874850396460804, 6.64404285394586224726158658695, 7.56292576532902279117260788537, 7.916065177931350363735154918886, 8.465802402483926236510824704457, 8.595108553937894798103835727892, 9.621349777941533985981367704140, 9.701101184568594692011608215984, 10.34688336473285712867502854354, 10.62583505301402662394441528983, 11.41012945784341050027251651536, 12.23107327955428078505588443207
