| L(s) = 1 | − 8·2-s + 16·4-s + 17·5-s − 408·7-s + 128·8-s − 136·10-s + 145·11-s − 1.43e3·13-s + 3.26e3·14-s − 1.02e3·16-s + 1.37e3·17-s − 1.08e3·19-s + 272·20-s − 1.16e3·22-s − 4.50e3·23-s + 136·25-s + 1.14e4·26-s − 6.52e3·28-s − 1.57e4·29-s − 8.81e3·31-s + 2.04e3·32-s − 1.09e4·34-s − 6.93e3·35-s + 1.45e4·37-s + 8.64e3·38-s + 2.17e3·40-s − 1.47e4·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.304·5-s − 3.14·7-s + 0.707·8-s − 0.430·10-s + 0.361·11-s − 2.34·13-s + 4.45·14-s − 16-s + 1.15·17-s − 0.686·19-s + 0.152·20-s − 0.510·22-s − 1.77·23-s + 0.0435·25-s + 3.31·26-s − 1.57·28-s − 3.47·29-s − 1.64·31-s + 0.353·32-s − 1.62·34-s − 0.957·35-s + 1.75·37-s + 0.971·38-s + 0.215·40-s − 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.001619173422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001619173422\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 408 T + 1525 p^{2} T^{2} + 408 p^{5} T^{3} + p^{10} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 17 T + 153 T^{2} + 103938 T^{3} - 10650254 T^{4} + 103938 p^{5} T^{5} + 153 p^{10} T^{6} - 17 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 145 T - 3207 T^{2} + 43191150 T^{3} - 28736332052 T^{4} + 43191150 p^{5} T^{5} - 3207 p^{10} T^{6} - 145 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 55 p T + 864206 T^{2} + 55 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 1372 T + 1046574 T^{2} + 2749356288 T^{3} - 3990131643437 T^{4} + 2749356288 p^{5} T^{5} + 1046574 p^{10} T^{6} - 1372 p^{15} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 1081 T + 433999 T^{2} - 4559264516 T^{3} - 8484957649496 T^{4} - 4559264516 p^{5} T^{5} + 433999 p^{10} T^{6} + 1081 p^{15} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 196 p T + 2467842 T^{2} + 976381056 p T^{3} + 146546957639683 T^{4} + 976381056 p^{6} T^{5} + 2467842 p^{10} T^{6} + 196 p^{16} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 7865 T + 34952518 T^{2} + 7865 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8816 T + 2245595 T^{2} + 160609526544 T^{3} + 2651926936135064 T^{4} + 160609526544 p^{5} T^{5} + 2245595 p^{10} T^{6} + 8816 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 14573 T + 61179319 T^{2} - 182236764008 T^{3} + 3324004250276398 T^{4} - 182236764008 p^{5} T^{5} + 61179319 p^{10} T^{6} - 14573 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 7350 T + 218270722 T^{2} + 7350 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 5921 T + 278228220 T^{2} + 5921 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44808 T + 1055145014 T^{2} - 22131649627488 T^{3} + 394401192496214403 T^{4} - 22131649627488 p^{5} T^{5} + 1055145014 p^{10} T^{6} - 44808 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 9417 T - 757353913 T^{2} - 90806398272 T^{3} + 503800193861521878 T^{4} - 90806398272 p^{5} T^{5} - 757353913 p^{10} T^{6} - 9417 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 5077 T - 531271125 T^{2} - 4431213438888 T^{3} - 219243170901961256 T^{4} - 4431213438888 p^{5} T^{5} - 531271125 p^{10} T^{6} + 5077 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 42368 T - 290546614 T^{2} + 16794736040448 T^{3} + 2120269165248284219 T^{4} + 16794736040448 p^{5} T^{5} - 290546614 p^{10} T^{6} + 42368 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 30501 T - 1380866887 T^{2} - 11867095015326 T^{3} + 2262669185659976988 T^{4} - 11867095015326 p^{5} T^{5} - 1380866887 p^{10} T^{6} + 30501 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 91744 T + 5413284586 T^{2} + 91744 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 85665 T + 2935374389 T^{2} - 22013733392250 T^{3} - 123398919101099178 T^{4} - 22013733392250 p^{5} T^{5} + 2935374389 p^{10} T^{6} - 85665 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 94646 T + 3492510509 T^{2} - 65188188816186 T^{3} - 7162029708206797036 T^{4} - 65188188816186 p^{5} T^{5} + 3492510509 p^{10} T^{6} + 94646 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 33841 T + 8158439620 T^{2} + 33841 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 27558 T - 10411241470 T^{2} - 70712064288 T^{3} + 89361702837866877519 T^{4} - 70712064288 p^{5} T^{5} - 10411241470 p^{10} T^{6} - 27558 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 46671 T + 13111898848 T^{2} + 46671 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−9.163974356766606349095576709484, −8.636929913640058031111248047868, −8.636061792901877372954930632298, −7.82361842779169921778109808777, −7.64746807534321529744365381192, −7.37348520673812319245836023811, −7.33692287704796007917620396889, −7.05703568557933883028114257026, −6.59468528851882931643378055098, −6.09545841723424166249682697723, −6.06656469198603740537732361099, −5.62087158157570828410379411993, −5.51613774677841542457579427657, −4.99610549546679328113440342182, −4.34125747588434945122476408127, −3.97137665213010938561102894025, −3.85252953625816693275740835354, −3.41243001700951834867657609749, −2.92631746521163932587182188917, −2.37727974951136362810315832611, −2.37556522626763676123950525099, −1.55911043271772239855648179807, −1.23024504944540698994645558709, −0.093291468246802901078975636956, −0.07386572554218455793732603621,
0.07386572554218455793732603621, 0.093291468246802901078975636956, 1.23024504944540698994645558709, 1.55911043271772239855648179807, 2.37556522626763676123950525099, 2.37727974951136362810315832611, 2.92631746521163932587182188917, 3.41243001700951834867657609749, 3.85252953625816693275740835354, 3.97137665213010938561102894025, 4.34125747588434945122476408127, 4.99610549546679328113440342182, 5.51613774677841542457579427657, 5.62087158157570828410379411993, 6.06656469198603740537732361099, 6.09545841723424166249682697723, 6.59468528851882931643378055098, 7.05703568557933883028114257026, 7.33692287704796007917620396889, 7.37348520673812319245836023811, 7.64746807534321529744365381192, 7.82361842779169921778109808777, 8.636061792901877372954930632298, 8.636929913640058031111248047868, 9.163974356766606349095576709484
