| L(s) = 1 | − 3·2-s + 4-s − 5·5-s − 22·7-s + 21·8-s + 15·10-s + 30·11-s − 46·13-s + 66·14-s − 71·16-s − 17·17-s + 104·19-s − 5·20-s − 90·22-s − 42·23-s + 25·25-s + 138·26-s − 22·28-s + 66·29-s + 194·31-s + 45·32-s + 51·34-s + 110·35-s + 206·37-s − 312·38-s − 105·40-s + 126·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.447·5-s − 1.18·7-s + 0.928·8-s + 0.474·10-s + 0.822·11-s − 0.981·13-s + 1.25·14-s − 1.10·16-s − 0.242·17-s + 1.25·19-s − 0.0559·20-s − 0.872·22-s − 0.380·23-s + 1/5·25-s + 1.04·26-s − 0.148·28-s + 0.422·29-s + 1.12·31-s + 0.248·32-s + 0.257·34-s + 0.531·35-s + 0.915·37-s − 1.33·38-s − 0.415·40-s + 0.479·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 17 | \( 1 + p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 194 T + p^{3} T^{2} \) |
| 37 | \( 1 - 206 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 - 540 T + p^{3} T^{2} \) |
| 53 | \( 1 + 78 T + p^{3} T^{2} \) |
| 59 | \( 1 + 432 T + p^{3} T^{2} \) |
| 61 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 848 T + p^{3} T^{2} \) |
| 71 | \( 1 - 174 T + p^{3} T^{2} \) |
| 73 | \( 1 - 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 398 T + p^{3} T^{2} \) |
| 83 | \( 1 + 828 T + p^{3} T^{2} \) |
| 89 | \( 1 + 630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1486 T + p^{3} T^{2} \) |
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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
−9.699324381824304205220792242421, −8.806424737046611671262755392553, −7.85302052434736521657383619886, −7.11784877146727825389121823919, −6.28697199858105103770440373133, −4.89131168175070856653163154550, −3.88867630720741266222431509401, −2.70150447400719831191527516516, −1.07658741977855915909604408345, 0,
1.07658741977855915909604408345, 2.70150447400719831191527516516, 3.88867630720741266222431509401, 4.89131168175070856653163154550, 6.28697199858105103770440373133, 7.11784877146727825389121823919, 7.85302052434736521657383619886, 8.806424737046611671262755392553, 9.699324381824304205220792242421
