| L(s) = 1 | + 3·2-s − 5·3-s + 4-s − 5·5-s − 15·6-s − 22·7-s − 21·8-s − 2·9-s − 15·10-s + 60·11-s − 5·12-s − 31·13-s − 66·14-s + 25·15-s − 71·16-s + 17·17-s − 6·18-s − 61·19-s − 5·20-s + 110·21-s + 180·22-s − 78·23-s + 105·24-s + 25·25-s − 93·26-s + 145·27-s − 22·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.962·3-s + 1/8·4-s − 0.447·5-s − 1.02·6-s − 1.18·7-s − 0.928·8-s − 0.0740·9-s − 0.474·10-s + 1.64·11-s − 0.120·12-s − 0.661·13-s − 1.25·14-s + 0.430·15-s − 1.10·16-s + 0.242·17-s − 0.0785·18-s − 0.736·19-s − 0.0559·20-s + 1.14·21-s + 1.74·22-s − 0.707·23-s + 0.893·24-s + 1/5·25-s − 0.701·26-s + 1.03·27-s − 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p T \) |
| 17 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 31 T + p^{3} T^{2} \) |
| 19 | \( 1 + 61 T + p^{3} T^{2} \) |
| 23 | \( 1 + 78 T + p^{3} T^{2} \) |
| 29 | \( 1 - 69 T + p^{3} T^{2} \) |
| 31 | \( 1 + p T + p^{3} T^{2} \) |
| 37 | \( 1 - 56 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 538 T + p^{3} T^{2} \) |
| 47 | \( 1 + 465 T + p^{3} T^{2} \) |
| 53 | \( 1 - 723 T + p^{3} T^{2} \) |
| 59 | \( 1 + 753 T + p^{3} T^{2} \) |
| 61 | \( 1 - 35 T + p^{3} T^{2} \) |
| 67 | \( 1 + 322 T + p^{3} T^{2} \) |
| 71 | \( 1 + 99 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1123 T + p^{3} T^{2} \) |
| 79 | \( 1 - 488 T + p^{3} T^{2} \) |
| 83 | \( 1 + 852 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1215 T + p^{3} T^{2} \) |
| 97 | \( 1 + 601 T + p^{3} T^{2} \) |
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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
−13.04913822005293449988417114682, −12.08879730859057078263436207930, −11.66541073414219267869149826465, −10.02688914032282294193474536700, −8.825587709825755242747556026354, −6.70404560290825159465418499310, −6.02935947675107418339717368790, −4.58799521159042287827864507737, −3.37411114716370851441639990928, 0,
3.37411114716370851441639990928, 4.58799521159042287827864507737, 6.02935947675107418339717368790, 6.70404560290825159465418499310, 8.825587709825755242747556026354, 10.02688914032282294193474536700, 11.66541073414219267869149826465, 12.08879730859057078263436207930, 13.04913822005293449988417114682
