| L(s) = 1 | − 0.381·3-s − 2.23·5-s − 7-s − 2.85·9-s − 3.85·11-s + 6.23·13-s + 0.854·15-s − 5.85·17-s + 19-s + 0.381·21-s − 1.76·23-s + 2.23·27-s + 6.61·29-s + 4.61·31-s + 1.47·33-s + 2.23·35-s + 7.47·37-s − 2.38·39-s + 9.56·41-s + 10.9·43-s + 6.38·45-s − 7·47-s + 49-s + 2.23·51-s − 4.85·53-s + 8.61·55-s − 0.381·57-s + ⋯ |
| L(s) = 1 | − 0.220·3-s − 0.999·5-s − 0.377·7-s − 0.951·9-s − 1.16·11-s + 1.72·13-s + 0.220·15-s − 1.41·17-s + 0.229·19-s + 0.0833·21-s − 0.367·23-s + 0.430·27-s + 1.22·29-s + 0.829·31-s + 0.256·33-s + 0.377·35-s + 1.22·37-s − 0.381·39-s + 1.49·41-s + 1.66·43-s + 0.951·45-s − 1.02·47-s + 0.142·49-s + 0.313·51-s − 0.666·53-s + 1.16·55-s − 0.0505·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 - 4.61T + 31T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 + 4.85T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 + 3.38T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.85T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 5.23T + 89T^{2} \) |
| 97 | \( 1 + 8.23T + 97T^{2} \) |
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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
−7.66888699307078912686885728545, −6.54576802768708709196431292023, −6.17705866010711341039418698433, −5.45171189485496904254715341867, −4.45816033775996793719805448029, −3.99484949987145261178538121382, −2.99078088527646222087092214049, −2.49753412461605911796078231868, −0.949636637073059463511183550724, 0,
0.949636637073059463511183550724, 2.49753412461605911796078231868, 2.99078088527646222087092214049, 3.99484949987145261178538121382, 4.45816033775996793719805448029, 5.45171189485496904254715341867, 6.17705866010711341039418698433, 6.54576802768708709196431292023, 7.66888699307078912686885728545
