| L(s) = 1 | + 27·3-s − 18·5-s − 343·7-s + 729·9-s − 8.17e3·11-s − 1.42e4·13-s − 486·15-s − 2.14e4·17-s + 5.88e3·19-s − 9.26e3·21-s + 9.87e4·23-s − 7.78e4·25-s + 1.96e4·27-s + 1.65e5·29-s + 2.41e5·31-s − 2.20e5·33-s + 6.17e3·35-s + 1.85e5·37-s − 3.84e5·39-s + 5.96e4·41-s + 8.09e5·43-s − 1.31e4·45-s − 9.42e5·47-s + 1.17e5·49-s − 5.79e5·51-s + 2.26e5·53-s + 1.47e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0643·5-s − 0.377·7-s + 1/3·9-s − 1.85·11-s − 1.79·13-s − 0.0371·15-s − 1.05·17-s + 0.196·19-s − 0.218·21-s + 1.69·23-s − 0.995·25-s + 0.192·27-s + 1.25·29-s + 1.45·31-s − 1.06·33-s + 0.0243·35-s + 0.601·37-s − 1.03·39-s + 0.135·41-s + 1.55·43-s − 0.0214·45-s − 1.32·47-s + 1/7·49-s − 0.611·51-s + 0.208·53-s + 0.119·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.574608791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.574608791\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| 7 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 + 18 T + p^{7} T^{2} \) |
| 11 | \( 1 + 8172 T + p^{7} T^{2} \) |
| 13 | \( 1 + 14242 T + p^{7} T^{2} \) |
| 17 | \( 1 + 21462 T + p^{7} T^{2} \) |
| 19 | \( 1 - 5884 T + p^{7} T^{2} \) |
| 23 | \( 1 - 98784 T + p^{7} T^{2} \) |
| 29 | \( 1 - 165174 T + p^{7} T^{2} \) |
| 31 | \( 1 - 241312 T + p^{7} T^{2} \) |
| 37 | \( 1 - 185438 T + p^{7} T^{2} \) |
| 41 | \( 1 - 59682 T + p^{7} T^{2} \) |
| 43 | \( 1 - 809308 T + p^{7} T^{2} \) |
| 47 | \( 1 + 942096 T + p^{7} T^{2} \) |
| 53 | \( 1 - 226398 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2205732 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1156690 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3740404 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2593296 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1038742 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2280032 T + p^{7} T^{2} \) |
| 83 | \( 1 - 283404 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5227230 T + p^{7} T^{2} \) |
| 97 | \( 1 - 6168770 T + p^{7} T^{2} \) |
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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
−10.15169264930751325884542090445, −9.560751493136091847202281625442, −8.373787111207332661564329029926, −7.58918404875320912701682763041, −6.73170219722989646104878986482, −5.24539060509774471326636344483, −4.49131744632304455943523559193, −2.80749075884772852403606027762, −2.45098324817472424885529743416, −0.54833022780874073047632440046,
0.54833022780874073047632440046, 2.45098324817472424885529743416, 2.80749075884772852403606027762, 4.49131744632304455943523559193, 5.24539060509774471326636344483, 6.73170219722989646104878986482, 7.58918404875320912701682763041, 8.373787111207332661564329029926, 9.560751493136091847202281625442, 10.15169264930751325884542090445
