| L(s) = 1 | + 3-s + 20·5-s + 10·7-s − 42·9-s − 30·11-s + 153·13-s + 20·15-s − 68·17-s − 120·19-s + 10·21-s + 92·23-s − 88·25-s − 37·27-s + 315·29-s − 249·31-s − 30·33-s + 200·35-s + 348·37-s + 153·39-s − 929·41-s + 50·43-s − 840·45-s − 205·47-s − 408·49-s − 68·51-s + 578·53-s − 600·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 1.78·5-s + 0.539·7-s − 1.55·9-s − 0.822·11-s + 3.26·13-s + 0.344·15-s − 0.970·17-s − 1.44·19-s + 0.103·21-s + 0.834·23-s − 0.703·25-s − 0.263·27-s + 2.01·29-s − 1.44·31-s − 0.158·33-s + 0.965·35-s + 1.54·37-s + 0.628·39-s − 3.53·41-s + 0.177·43-s − 2.78·45-s − 0.636·47-s − 1.18·49-s − 0.186·51-s + 1.49·53-s − 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.809203525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.809203525\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - T + 43 T^{2} - 16 p T^{3} + 296 p T^{4} - 16 p^{4} T^{5} + 43 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 4 p T + 488 T^{2} - 6996 T^{3} + 89886 T^{4} - 6996 p^{3} T^{5} + 488 p^{6} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 10 T + 508 T^{2} + 3142 T^{3} + 105542 T^{4} + 3142 p^{3} T^{5} + 508 p^{6} T^{6} - 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 30 T + 2792 T^{2} + 57870 T^{3} + 4873710 T^{4} + 57870 p^{3} T^{5} + 2792 p^{6} T^{6} + 30 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 153 T + 13253 T^{2} - 59898 p T^{3} + 38847186 T^{4} - 59898 p^{4} T^{5} + 13253 p^{6} T^{6} - 153 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 4 p T + 14184 T^{2} + 669060 T^{3} + 89571838 T^{4} + 669060 p^{3} T^{5} + 14184 p^{6} T^{6} + 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 120 T + 1612 p T^{2} + 2449368 T^{3} + 325044918 T^{4} + 2449368 p^{3} T^{5} + 1612 p^{7} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 315 T + 94165 T^{2} - 18517710 T^{3} + 3578199242 T^{4} - 18517710 p^{3} T^{5} + 94165 p^{6} T^{6} - 315 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 249 T + 25179 T^{2} - 7287372 T^{3} - 1985430632 T^{4} - 7287372 p^{3} T^{5} + 25179 p^{6} T^{6} + 249 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 348 T + 188200 T^{2} - 39268892 T^{3} + 12974765214 T^{4} - 39268892 p^{3} T^{5} + 188200 p^{6} T^{6} - 348 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 929 T + 359381 T^{2} + 69061038 T^{3} + 11316059274 T^{4} + 69061038 p^{3} T^{5} + 359381 p^{6} T^{6} + 929 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 50 T + 87932 T^{2} - 15376674 T^{3} + 3065269142 T^{4} - 15376674 p^{3} T^{5} + 87932 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 205 T + 308899 T^{2} + 65990732 T^{3} + 43468201592 T^{4} + 65990732 p^{3} T^{5} + 308899 p^{6} T^{6} + 205 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 578 T + 281992 T^{2} - 85871110 T^{3} + 41753084318 T^{4} - 85871110 p^{3} T^{5} + 281992 p^{6} T^{6} - 578 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 664 T + 217628 T^{2} - 56500072 T^{3} - 48154855786 T^{4} - 56500072 p^{3} T^{5} + 217628 p^{6} T^{6} + 664 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 182 T + 839632 T^{2} - 122764402 T^{3} + 278181229358 T^{4} - 122764402 p^{3} T^{5} + 839632 p^{6} T^{6} - 182 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1078 T + 1377944 T^{2} - 910974966 T^{3} + 649636246382 T^{4} - 910974966 p^{3} T^{5} + 1377944 p^{6} T^{6} - 1078 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 739 T + 1008315 T^{2} - 449566116 T^{3} + 417304081672 T^{4} - 449566116 p^{3} T^{5} + 1008315 p^{6} T^{6} - 739 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 921 T + 968409 T^{2} - 372977286 T^{3} + 304024198294 T^{4} - 372977286 p^{3} T^{5} + 968409 p^{6} T^{6} - 921 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 1754 T + 2596768 T^{2} + 2500649610 T^{3} + 2049150508446 T^{4} + 2500649610 p^{3} T^{5} + 2596768 p^{6} T^{6} + 1754 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1020 T + 2228264 T^{2} - 1583381380 T^{3} + 1909520366862 T^{4} - 1583381380 p^{3} T^{5} + 2228264 p^{6} T^{6} - 1020 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 430 T + 1361892 T^{2} + 334282362 T^{3} + 1199003231206 T^{4} + 334282362 p^{3} T^{5} + 1361892 p^{6} T^{6} + 430 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 48 T + 1786672 T^{2} + 667882152 T^{3} + 1735933824846 T^{4} + 667882152 p^{3} T^{5} + 1786672 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−6.42939238021668107673504958052, −6.11912177283485649586456308534, −6.07283462517296250855576113434, −5.74530731805368169213196940810, −5.63096937141379552815794724682, −5.18828726412000153136090200568, −5.14896731288608412666787565558, −5.06749204524607896649303798313, −4.85987588093021200043826591492, −4.10194290510178793733911784444, −4.07948728110511268267490364783, −4.03269682935774204073371397074, −3.78554279916467140535528337516, −3.19493439463206085987331740582, −3.19393518710012290360568154097, −2.83239260500124020672964473612, −2.69459725071902991609383646993, −2.32507268742733253954946052997, −1.92081108205958589741405785566, −1.76935253926689321816642774912, −1.68381061662655698593117081767, −1.44019368669449011070696046642, −0.800444164613185846375939405987, −0.52320242245184258425608438634, −0.33982794010532946786805569606,
0.33982794010532946786805569606, 0.52320242245184258425608438634, 0.800444164613185846375939405987, 1.44019368669449011070696046642, 1.68381061662655698593117081767, 1.76935253926689321816642774912, 1.92081108205958589741405785566, 2.32507268742733253954946052997, 2.69459725071902991609383646993, 2.83239260500124020672964473612, 3.19393518710012290360568154097, 3.19493439463206085987331740582, 3.78554279916467140535528337516, 4.03269682935774204073371397074, 4.07948728110511268267490364783, 4.10194290510178793733911784444, 4.85987588093021200043826591492, 5.06749204524607896649303798313, 5.14896731288608412666787565558, 5.18828726412000153136090200568, 5.63096937141379552815794724682, 5.74530731805368169213196940810, 6.07283462517296250855576113434, 6.11912177283485649586456308534, 6.42939238021668107673504958052
