| L(s) = 1 | + 4·2-s + 4·3-s + 16·4-s − 9·5-s + 16·6-s − 32·7-s + 64·8-s − 227·9-s − 36·10-s + 64·12-s + 145·13-s − 128·14-s − 36·15-s + 256·16-s − 603·17-s − 908·18-s − 1.44e3·19-s − 144·20-s − 128·21-s − 60·23-s + 256·24-s − 3.04e3·25-s + 580·26-s − 1.88e3·27-s − 512·28-s − 4.40e3·29-s − 144·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.256·3-s + 1/2·4-s − 0.160·5-s + 0.181·6-s − 0.246·7-s + 0.353·8-s − 0.934·9-s − 0.113·10-s + 0.128·12-s + 0.237·13-s − 0.174·14-s − 0.0413·15-s + 1/4·16-s − 0.506·17-s − 0.660·18-s − 0.920·19-s − 0.0804·20-s − 0.0633·21-s − 0.0236·23-s + 0.0907·24-s − 0.974·25-s + 0.168·26-s − 0.496·27-s − 0.123·28-s − 0.973·29-s − 0.0292·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{5} T^{2} \) |
| 5 | \( 1 + 9 T + p^{5} T^{2} \) |
| 7 | \( 1 + 32 T + p^{5} T^{2} \) |
| 13 | \( 1 - 145 T + p^{5} T^{2} \) |
| 17 | \( 1 + 603 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1448 T + p^{5} T^{2} \) |
| 23 | \( 1 + 60 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4407 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2104 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13055 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1215 T + p^{5} T^{2} \) |
| 43 | \( 1 + 6476 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24300 T + p^{5} T^{2} \) |
| 53 | \( 1 - 12363 T + p^{5} T^{2} \) |
| 59 | \( 1 - 32340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 42782 T + p^{5} T^{2} \) |
| 67 | \( 1 - 56288 T + p^{5} T^{2} \) |
| 71 | \( 1 + 54084 T + p^{5} T^{2} \) |
| 73 | \( 1 + 16394 T + p^{5} T^{2} \) |
| 79 | \( 1 + 76700 T + p^{5} T^{2} \) |
| 83 | \( 1 + 71928 T + p^{5} T^{2} \) |
| 89 | \( 1 - 97539 T + p^{5} T^{2} \) |
| 97 | \( 1 - 93467 T + p^{5} T^{2} \) |
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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
−11.11997587993662993266444567918, −9.844735952686107402632223905412, −8.716995582570508607463395468855, −7.77724218109066697105712605791, −6.50096596863013815461932658732, −5.64135206842661680981282161994, −4.30837534390478344924934362080, −3.22945546412563708943395523096, −2.02024935682968467477326012861, 0,
2.02024935682968467477326012861, 3.22945546412563708943395523096, 4.30837534390478344924934362080, 5.64135206842661680981282161994, 6.50096596863013815461932658732, 7.77724218109066697105712605791, 8.716995582570508607463395468855, 9.844735952686107402632223905412, 11.11997587993662993266444567918
