| L(s) = 1 | + 94·3-s − 330·5-s + 686·7-s + 3.11e3·9-s + 2.84e3·11-s − 2.53e3·13-s − 3.10e4·15-s − 1.48e3·17-s + 3.28e4·19-s + 6.44e4·21-s + 6.57e3·23-s − 5.29e4·25-s − 3.88e4·27-s − 2.06e4·29-s + 3.91e5·31-s + 2.67e5·33-s − 2.26e5·35-s − 3.67e5·37-s − 2.38e5·39-s + 7.34e5·41-s − 4.80e5·43-s − 1.02e6·45-s + 1.08e6·47-s + 3.52e5·49-s − 1.39e5·51-s − 2.85e6·53-s − 9.38e5·55-s + ⋯ |
| L(s) = 1 | + 2.01·3-s − 1.18·5-s + 0.755·7-s + 1.42·9-s + 0.644·11-s − 0.319·13-s − 2.37·15-s − 0.0734·17-s + 1.09·19-s + 1.51·21-s + 0.112·23-s − 0.677·25-s − 0.379·27-s − 0.157·29-s + 2.36·31-s + 1.29·33-s − 0.892·35-s − 1.19·37-s − 0.642·39-s + 1.66·41-s − 0.921·43-s − 1.68·45-s + 1.53·47-s + 3/7·49-s − 0.147·51-s − 2.63·53-s − 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.264767494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.264767494\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 94 T + 1906 p T^{2} - 94 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 66 p T + 6474 p^{2} T^{2} + 66 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2844 T + 38086566 T^{2} - 2844 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2534 T - 41123742 T^{2} + 2534 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1488 T + 798529822 T^{2} + 1488 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32810 T + 1897672038 T^{2} - 32810 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6576 T + 6819963598 T^{2} - 6576 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 20640 T + 15579628518 T^{2} + 20640 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 391836 T + 92048864606 T^{2} - 391836 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 367392 T + 63852768182 T^{2} + 367392 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 734664 T + 402811824126 T^{2} - 734664 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 480476 T + 594501933318 T^{2} + 480476 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1089108 T + 1015337137342 T^{2} - 1089108 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2858844 T + 4386858062398 T^{2} + 2858844 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 160170 T + 4361928868198 T^{2} - 160170 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 864646 T + 5755969170906 T^{2} - 864646 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 328648 T + 11587546356582 T^{2} + 328648 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7500216 T + 28549732695406 T^{2} - 7500216 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4301244 T + 18754109784038 T^{2} - 4301244 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6408440 T + 32072611946718 T^{2} - 6408440 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11659074 T + 84453675852838 T^{2} - 11659074 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9772260 T + 83812995056598 T^{2} - 9772260 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10762752 T + 188617737573662 T^{2} - 10762752 p^{7} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.759904373269112988955124498899, −9.608878340826532956144945091352, −9.253023775063838874700766631258, −8.575503265899280530384091635441, −8.360857886950874978677600511717, −7.964758971798018543541135494696, −7.55634526624300242108277628847, −7.44835633870576155781046270782, −6.50816089850573483209892494852, −6.17313918717515513481778629212, −5.17518600750258263121838481886, −4.85917205175251389431176452007, −4.16026492989531971266560595061, −3.74088856273546613172812860380, −3.31865398545849082423369531626, −2.92293892113398593119207360759, −2.05378236009486930510261385339, −2.01839175185472348385462499920, −0.870271511632722298562187943072, −0.60721144106077162679387462478,
0.60721144106077162679387462478, 0.870271511632722298562187943072, 2.01839175185472348385462499920, 2.05378236009486930510261385339, 2.92293892113398593119207360759, 3.31865398545849082423369531626, 3.74088856273546613172812860380, 4.16026492989531971266560595061, 4.85917205175251389431176452007, 5.17518600750258263121838481886, 6.17313918717515513481778629212, 6.50816089850573483209892494852, 7.44835633870576155781046270782, 7.55634526624300242108277628847, 7.964758971798018543541135494696, 8.360857886950874978677600511717, 8.575503265899280530384091635441, 9.253023775063838874700766631258, 9.608878340826532956144945091352, 9.759904373269112988955124498899
