| L(s) = 1 | + 256·4-s − 239·7-s + 4.16e5·19-s − 3.90e5·25-s − 6.11e4·28-s + 2.39e6·31-s + 2.96e6·37-s + 1.36e7·43-s − 5.70e6·49-s + 2.44e7·61-s − 1.67e7·64-s − 3.72e7·67-s − 2.28e7·73-s + 1.06e8·76-s − 7.48e7·79-s − 1.00e8·100-s − 1.24e8·103-s + 6.81e7·109-s + 2.14e8·121-s + 6.12e8·124-s + 127-s + 131-s − 9.95e7·133-s + 137-s + 139-s + 7.59e8·148-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s − 0.0995·7-s + 3.19·19-s − 25-s − 0.0995·28-s + 2.59·31-s + 1.58·37-s + 3.99·43-s − 0.990·49-s + 1.76·61-s − 64-s − 1.85·67-s − 0.806·73-s + 3.19·76-s − 1.92·79-s − 100-s − 1.10·103-s + 0.482·109-s + 121-s + 2.59·124-s − 0.318·133-s + 1.58·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(4.471349040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.471349040\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 239 T + p^{8} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p^{8} T^{2} + p^{16} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{8} T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 20641 T + p^{8} T^{2} )( 1 + 20641 T + p^{8} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 258526 T + p^{8} T^{2} )( 1 - 157967 T + p^{8} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{8} T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1809406 T + p^{8} T^{2} )( 1 - 583439 T + p^{8} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3468481 T + p^{8} T^{2} )( 1 + 503522 T + p^{8} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6837073 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - p^{8} T^{2} + p^{16} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 24133919 T + p^{8} T^{2} )( 1 - 307393 T + p^{8} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5421406 T + p^{8} T^{2} )( 1 + 31874833 T + p^{8} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16169282 T + p^{8} T^{2} )( 1 + 39067199 T + p^{8} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 18887038 T + p^{8} T^{2} )( 1 + 56007121 T + p^{8} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 176908034 T + p^{8} T^{2} )( 1 + 176908034 T + p^{8} T^{2} ) \) |
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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
−13.59572749848923047568903525513, −13.06692606316925045388354696752, −12.12926501204130282483349543234, −11.85449822988354180504757393619, −11.39546102615457504936321851721, −10.89957991634720377388219801685, −9.883444173507441036606665170353, −9.772191031526930645740631155698, −9.005945150905648976367180811473, −8.024240275163906909555281840409, −7.47653957146487927667984132246, −7.16191837158659595587005149316, −6.04911246902548639209313030173, −5.86896899055920695320111578469, −4.80667640793934608136129210346, −4.00101093707779898749128625119, −2.81810976858910653398645799182, −2.66796737970494788354670329687, −1.32242272933818938675535662222, −0.77371123765152503683173857284,
0.77371123765152503683173857284, 1.32242272933818938675535662222, 2.66796737970494788354670329687, 2.81810976858910653398645799182, 4.00101093707779898749128625119, 4.80667640793934608136129210346, 5.86896899055920695320111578469, 6.04911246902548639209313030173, 7.16191837158659595587005149316, 7.47653957146487927667984132246, 8.024240275163906909555281840409, 9.005945150905648976367180811473, 9.772191031526930645740631155698, 9.883444173507441036606665170353, 10.89957991634720377388219801685, 11.39546102615457504936321851721, 11.85449822988354180504757393619, 12.12926501204130282483349543234, 13.06692606316925045388354696752, 13.59572749848923047568903525513
