| L(s) = 1 | + 4·2-s + 16·4-s + 118·7-s + 64·8-s − 192·11-s − 1.10e3·13-s + 472·14-s + 256·16-s + 762·17-s − 2.74e3·19-s − 768·22-s + 1.56e3·23-s − 4.42e3·26-s + 1.88e3·28-s − 5.91e3·29-s − 6.86e3·31-s + 1.02e3·32-s + 3.04e3·34-s + 5.51e3·37-s − 1.09e4·38-s + 378·41-s + 2.43e3·43-s − 3.07e3·44-s + 6.26e3·46-s + 1.31e4·47-s − 2.88e3·49-s − 1.76e4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.910·7-s + 0.353·8-s − 0.478·11-s − 1.81·13-s + 0.643·14-s + 1/4·16-s + 0.639·17-s − 1.74·19-s − 0.338·22-s + 0.617·23-s − 1.28·26-s + 0.455·28-s − 1.30·29-s − 1.28·31-s + 0.176·32-s + 0.452·34-s + 0.662·37-s − 1.23·38-s + 0.0351·41-s + 0.200·43-s − 0.239·44-s + 0.436·46-s + 0.866·47-s − 0.171·49-s − 0.907·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 118 T + p^{5} T^{2} \) |
| 11 | \( 1 + 192 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1106 T + p^{5} T^{2} \) |
| 17 | \( 1 - 762 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2740 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1566 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5910 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6868 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5518 T + p^{5} T^{2} \) |
| 41 | \( 1 - 378 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2434 T + p^{5} T^{2} \) |
| 47 | \( 1 - 13122 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9174 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34980 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9838 T + p^{5} T^{2} \) |
| 67 | \( 1 + 33722 T + p^{5} T^{2} \) |
| 71 | \( 1 + 70212 T + p^{5} T^{2} \) |
| 73 | \( 1 + 21986 T + p^{5} T^{2} \) |
| 79 | \( 1 - 4520 T + p^{5} T^{2} \) |
| 83 | \( 1 + 109074 T + p^{5} T^{2} \) |
| 89 | \( 1 + 38490 T + p^{5} T^{2} \) |
| 97 | \( 1 - 1918 T + p^{5} T^{2} \) |
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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
−10.03703629388111999753359620618, −8.885352449420041893953366594059, −7.72431319257830898938131380685, −7.17851145003773660209544620406, −5.79037555069095401169705758944, −4.98195131267043482520231747877, −4.15417989813524516205224832000, −2.69627715918362215975567392012, −1.78399724921995736555779970846, 0,
1.78399724921995736555779970846, 2.69627715918362215975567392012, 4.15417989813524516205224832000, 4.98195131267043482520231747877, 5.79037555069095401169705758944, 7.17851145003773660209544620406, 7.72431319257830898938131380685, 8.885352449420041893953366594059, 10.03703629388111999753359620618
