| L(s) = 1 | + 1.03·3-s − 5-s + 2.60·7-s − 1.92·9-s − 2.87·13-s − 1.03·15-s + 0.810·17-s − 5.15·19-s + 2.70·21-s − 4.98·23-s + 25-s − 5.10·27-s + 6.72·29-s + 4.14·31-s − 2.60·35-s + 4.05·37-s − 2.98·39-s + 9.76·41-s + 5.92·43-s + 1.92·45-s + 0.967·47-s − 0.229·49-s + 0.841·51-s + 4.47·53-s − 5.34·57-s − 6.80·59-s − 2.61·61-s + ⋯ |
| L(s) = 1 | + 0.599·3-s − 0.447·5-s + 0.983·7-s − 0.640·9-s − 0.798·13-s − 0.267·15-s + 0.196·17-s − 1.18·19-s + 0.589·21-s − 1.03·23-s + 0.200·25-s − 0.983·27-s + 1.24·29-s + 0.743·31-s − 0.439·35-s + 0.666·37-s − 0.478·39-s + 1.52·41-s + 0.904·43-s + 0.286·45-s + 0.141·47-s − 0.0327·49-s + 0.117·51-s + 0.614·53-s − 0.708·57-s − 0.886·59-s − 0.334·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.03T + 3T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 - 0.810T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 + 4.98T + 23T^{2} \) |
| 29 | \( 1 - 6.72T + 29T^{2} \) |
| 31 | \( 1 - 4.14T + 31T^{2} \) |
| 37 | \( 1 - 4.05T + 37T^{2} \) |
| 41 | \( 1 - 9.76T + 41T^{2} \) |
| 43 | \( 1 - 5.92T + 43T^{2} \) |
| 47 | \( 1 - 0.967T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 6.80T + 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 + 7.96T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 2.99T + 97T^{2} \) |
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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
−7.66296190544247081353812494007, −6.70933487194607272343470709790, −5.97018060993344905137861910354, −5.22795834800411174841640783302, −4.33824260125442420223539463132, −4.04450741684134555693975080008, −2.70122246154123350308911439396, −2.48241434776316694967587147448, −1.29635588919958117832844519393, 0,
1.29635588919958117832844519393, 2.48241434776316694967587147448, 2.70122246154123350308911439396, 4.04450741684134555693975080008, 4.33824260125442420223539463132, 5.22795834800411174841640783302, 5.97018060993344905137861910354, 6.70933487194607272343470709790, 7.66296190544247081353812494007
