| L(s) = 1 | + 4·2-s + 12·4-s − 12·5-s + 32·8-s − 48·10-s + 4·11-s + 48·13-s + 80·16-s − 84·17-s − 72·19-s − 144·20-s + 16·22-s + 308·23-s − 44·25-s + 192·26-s + 80·29-s + 384·31-s + 192·32-s − 336·34-s + 536·37-s − 288·38-s − 384·40-s − 756·41-s + 400·43-s + 48·44-s + 1.23e3·46-s − 312·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.07·5-s + 1.41·8-s − 1.51·10-s + 0.109·11-s + 1.02·13-s + 5/4·16-s − 1.19·17-s − 0.869·19-s − 1.60·20-s + 0.155·22-s + 2.79·23-s − 0.351·25-s + 1.44·26-s + 0.512·29-s + 2.22·31-s + 1.06·32-s − 1.69·34-s + 2.38·37-s − 1.22·38-s − 1.51·40-s − 2.87·41-s + 1.41·43-s + 0.164·44-s + 3.94·46-s − 0.968·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.030431242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.030431242\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 12 T + 188 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T - 862 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 48 T + 4088 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 84 T + 676 p T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 72 T + 14622 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 308 T + 44522 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 80 T + 18626 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 384 T + 92918 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 536 T + 159018 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 756 T + 278276 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 400 T + 142566 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 312 T + 222182 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 52 T + 241982 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 864 T + 596990 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1416 T + 938664 T^{2} - 1416 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 144 T + 550262 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1524 T + 1208266 T^{2} - 1524 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 744 T + 241296 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 976 T + 532734 T^{2} - 976 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 312 T - 644698 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 108 T + 1330436 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 744 T + 432480 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.981487478041831987089235243144, −9.673909963371567744027155876366, −8.982511517731042136714813613000, −8.573191827183235233810501577268, −8.077141912333788968308416159450, −8.000464215803800459443691515934, −7.18949075342023757223628090172, −6.74607890087432979379391441848, −6.37923351868109786886332290783, −6.34833181977444668783065387084, −5.40297935990100448087521989103, −4.90489864555221227741914496199, −4.62381056340060406599873012446, −4.18504720589276411828160448098, −3.60816492098508621445407491030, −3.29744165442864350928071707646, −2.58429822285273829867433351551, −2.17277254734488591049556046602, −1.09777607819790063697322233089, −0.66207417989178348986228028054,
0.66207417989178348986228028054, 1.09777607819790063697322233089, 2.17277254734488591049556046602, 2.58429822285273829867433351551, 3.29744165442864350928071707646, 3.60816492098508621445407491030, 4.18504720589276411828160448098, 4.62381056340060406599873012446, 4.90489864555221227741914496199, 5.40297935990100448087521989103, 6.34833181977444668783065387084, 6.37923351868109786886332290783, 6.74607890087432979379391441848, 7.18949075342023757223628090172, 8.000464215803800459443691515934, 8.077141912333788968308416159450, 8.573191827183235233810501577268, 8.982511517731042136714813613000, 9.673909963371567744027155876366, 9.981487478041831987089235243144
