| L(s) = 1 | − 16·2-s + 256·4-s − 4.09e3·8-s + 2.71e4·11-s + 6.55e4·16-s − 1.62e5·17-s − 7.22e4·19-s − 4.34e5·22-s + 3.90e5·25-s − 1.04e6·32-s + 2.59e6·34-s + 1.15e6·38-s + 4.09e6·41-s + 5.42e6·43-s + 6.95e6·44-s + 5.76e6·49-s − 6.25e6·50-s + 2.41e7·59-s + 1.67e7·64-s − 1.39e7·67-s − 4.15e7·68-s + 3.35e7·73-s − 1.85e7·76-s − 6.55e7·82-s − 3.02e7·83-s − 8.68e7·86-s − 1.11e8·88-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 1.85·11-s + 16-s − 1.94·17-s − 0.554·19-s − 1.85·22-s + 25-s − 32-s + 1.94·34-s + 0.554·38-s + 1.45·41-s + 1.58·43-s + 1.85·44-s + 49-s − 50-s + 1.99·59-s + 64-s − 0.691·67-s − 1.94·68-s + 1.18·73-s − 0.554·76-s − 1.45·82-s − 0.636·83-s − 1.58·86-s − 1.85·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.287009537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287009537\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 11 | \( 1 - 27166 T + p^{8} T^{2} \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( 1 + 162434 T + p^{8} T^{2} \) |
| 19 | \( 1 + 72286 T + p^{8} T^{2} \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( 1 - 4099006 T + p^{8} T^{2} \) |
| 43 | \( 1 - 5426402 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( 1 - 24178078 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( 1 + 13944286 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 - 33567554 T + p^{8} T^{2} \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( 1 + 30209954 T + p^{8} T^{2} \) |
| 89 | \( 1 - 95519806 T + p^{8} T^{2} \) |
| 97 | \( 1 + 77418238 T + p^{8} T^{2} \) |
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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
−12.71652748465689903955636899162, −11.55855434625550147749629176946, −10.74347228709713992118239296042, −9.267792989298159224121322456115, −8.710847794623587356398636636618, −7.07369133049232427194012281561, −6.24796730326721899526532144922, −4.11622458808733833496086710795, −2.27506294062539411504738469965, −0.845024508693897854125912822917,
0.845024508693897854125912822917, 2.27506294062539411504738469965, 4.11622458808733833496086710795, 6.24796730326721899526532144922, 7.07369133049232427194012281561, 8.710847794623587356398636636618, 9.267792989298159224121322456115, 10.74347228709713992118239296042, 11.55855434625550147749629176946, 12.71652748465689903955636899162
