| L(s) = 1 | + 24·3-s − 14·4-s − 260·5-s + 534·9-s + 3.22e3·11-s − 336·12-s − 6.24e3·15-s − 6.50e3·16-s + 3.64e3·20-s + 1.14e4·23-s − 2.02e4·25-s + 2.80e4·27-s − 5.51e3·31-s + 7.73e4·33-s − 7.47e3·36-s + 7.60e4·37-s − 4.51e4·44-s − 1.38e5·45-s − 1.48e5·47-s − 1.56e5·48-s + 7.75e4·49-s + 2.14e5·53-s − 8.38e5·55-s + 7.66e5·59-s + 8.73e4·60-s + 1.27e5·64-s + 8.81e5·67-s + ⋯ |
| L(s) = 1 | + 8/9·3-s − 0.218·4-s − 2.07·5-s + 0.732·9-s + 2.42·11-s − 0.194·12-s − 1.84·15-s − 1.58·16-s + 0.454·20-s + 0.942·23-s − 1.29·25-s + 1.42·27-s − 0.185·31-s + 2.15·33-s − 0.160·36-s + 1.50·37-s − 0.529·44-s − 1.52·45-s − 1.42·47-s − 1.41·48-s + 0.659·49-s + 1.43·53-s − 5.03·55-s + 3.73·59-s + 0.404·60-s + 0.485·64-s + 2.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.764829447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.764829447\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $D_{4}$ | \( 1 - 3224 T + 42126 p^{2} T^{2} - 3224 p^{6} T^{3} + p^{12} T^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + 7 p T^{2} + 837 p^{3} T^{4} + 7 p^{13} T^{6} + p^{24} T^{8} \) |
| 3 | $D_{4}$ | \( ( 1 - 4 p T - 17 p T^{2} - 4 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 13 p T + p^{6} T^{2} )^{4} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 77596 T^{2} + 29113081206 T^{4} - 77596 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 13816636 T^{2} + 93901216948566 T^{4} - 13816636 p^{12} T^{6} + p^{24} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 27861916 T^{2} + 760434236368566 T^{4} - 27861916 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 123555964 T^{2} + 8195809477133526 T^{4} - 123555964 p^{12} T^{6} + p^{24} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 5732 T + 12077043 p T^{2} - 5732 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 870136804 T^{2} + 331303523963436966 T^{4} - 870136804 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 2756 T + 1097772141 T^{2} + 2756 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 38038 T + 3985032699 T^{2} - 38038 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 12542198044 T^{2} + 84437122435202446326 T^{4} - 12542198044 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 19351118596 T^{2} + \)\(16\!\cdots\!06\)\( T^{4} - 19351118596 p^{12} T^{6} + p^{24} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 74152 T + 22171678254 T^{2} + 74152 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 107132 T + 38269149894 T^{2} - 107132 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 383396 T + 118230507861 T^{2} - 383396 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 20914146244 T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - 20914146244 p^{12} T^{6} + p^{24} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 440908 T + 110550764829 T^{2} - 440908 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 640484 T + 274219890501 T^{2} - 640484 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 385025749276 T^{2} + 15534109173897189654 p^{2} T^{4} - 385025749276 p^{12} T^{6} + p^{24} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 524633940484 T^{2} + \)\(13\!\cdots\!26\)\( T^{4} - 524633940484 p^{12} T^{6} + p^{24} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 225527160676 T^{2} + \)\(19\!\cdots\!46\)\( T^{4} - 225527160676 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1245634 T + 1295235224931 T^{2} + 1245634 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 740038 T + 1356142925499 T^{2} - 740038 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.35117475513404405517141075553, −14.14959079399244077528508439673, −13.76608235632759314587610914813, −13.26279677149103133847424653477, −12.82574703033809718506823910392, −12.37263618765014976819498768840, −11.86504409786937939743486349426, −11.42280825632803170926257801972, −11.39441788263263646241601873744, −11.18041966463484002939450324144, −9.932869186035433691262929979275, −9.683104625067795493652529315538, −9.389924850115517639743184634030, −8.552750700696385249842188856656, −8.491102520163906718561984399566, −7.997795663058631022282608734142, −7.27671674023255377666670827720, −6.85195524308805392637980479411, −6.56197474483093495109337202258, −5.35439919678027486900636653250, −4.12557076219064349842380878188, −4.10955210006667715728463915789, −3.68136082069309228399813937632, −2.24498867069551432134592807796, −0.73902834143273722525451853712,
0.73902834143273722525451853712, 2.24498867069551432134592807796, 3.68136082069309228399813937632, 4.10955210006667715728463915789, 4.12557076219064349842380878188, 5.35439919678027486900636653250, 6.56197474483093495109337202258, 6.85195524308805392637980479411, 7.27671674023255377666670827720, 7.997795663058631022282608734142, 8.491102520163906718561984399566, 8.552750700696385249842188856656, 9.389924850115517639743184634030, 9.683104625067795493652529315538, 9.932869186035433691262929979275, 11.18041966463484002939450324144, 11.39441788263263646241601873744, 11.42280825632803170926257801972, 11.86504409786937939743486349426, 12.37263618765014976819498768840, 12.82574703033809718506823910392, 13.26279677149103133847424653477, 13.76608235632759314587610914813, 14.14959079399244077528508439673, 14.35117475513404405517141075553
