| L(s) = 1 | + 27·3-s − 378·5-s + 832·7-s + 729·9-s + 2.48e3·11-s + 1.48e4·13-s − 1.02e4·15-s − 2.23e4·17-s + 1.63e4·19-s + 2.24e4·21-s + 1.15e5·23-s + 6.47e4·25-s + 1.96e4·27-s + 1.57e5·29-s + 1.64e4·31-s + 6.70e4·33-s − 3.14e5·35-s − 1.49e5·37-s + 4.01e5·39-s − 2.41e5·41-s + 4.43e5·43-s − 2.75e5·45-s − 9.22e5·47-s − 1.31e5·49-s − 6.02e5·51-s − 6.97e5·53-s − 9.38e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.35·5-s + 0.916·7-s + 1/3·9-s + 0.562·11-s + 1.87·13-s − 0.780·15-s − 1.10·17-s + 0.545·19-s + 0.529·21-s + 1.97·23-s + 0.828·25-s + 0.192·27-s + 1.19·29-s + 0.0992·31-s + 0.324·33-s − 1.23·35-s − 0.484·37-s + 1.08·39-s − 0.546·41-s + 0.850·43-s − 0.450·45-s − 1.29·47-s − 0.159·49-s − 0.635·51-s − 0.643·53-s − 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.137904730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137904730\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 378 T + p^{7} T^{2} \) |
| 7 | \( 1 - 832 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2484 T + p^{7} T^{2} \) |
| 13 | \( 1 - 14870 T + p^{7} T^{2} \) |
| 17 | \( 1 + 22302 T + p^{7} T^{2} \) |
| 19 | \( 1 - 16300 T + p^{7} T^{2} \) |
| 23 | \( 1 - 115128 T + p^{7} T^{2} \) |
| 29 | \( 1 - 157086 T + p^{7} T^{2} \) |
| 31 | \( 1 - 16456 T + p^{7} T^{2} \) |
| 37 | \( 1 + 149266 T + p^{7} T^{2} \) |
| 41 | \( 1 + 241110 T + p^{7} T^{2} \) |
| 43 | \( 1 - 443188 T + p^{7} T^{2} \) |
| 47 | \( 1 + 922752 T + p^{7} T^{2} \) |
| 53 | \( 1 + 697626 T + p^{7} T^{2} \) |
| 59 | \( 1 + 870156 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2067062 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1680748 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1070280 T + p^{7} T^{2} \) |
| 73 | \( 1 + 2403334 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2301512 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4708692 T + p^{7} T^{2} \) |
| 89 | \( 1 - 4143690 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1622974 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.26843504201038771729272299861, −13.09919264664462661155473881128, −11.57809653585810852284140244944, −10.96840920719169677095147305925, −8.869660740028790614217460295381, −8.182856900749388794351069136987, −6.81903846397007326622668331961, −4.59902921586587528443435539192, −3.39825324640647508634544063135, −1.17215536804389033092036363236,
1.17215536804389033092036363236, 3.39825324640647508634544063135, 4.59902921586587528443435539192, 6.81903846397007326622668331961, 8.182856900749388794351069136987, 8.869660740028790614217460295381, 10.96840920719169677095147305925, 11.57809653585810852284140244944, 13.09919264664462661155473881128, 14.26843504201038771729272299861
