| L(s) = 1 | − 3-s − 2.09·5-s − 2.97·7-s + 9-s + 0.138·11-s + 5.06·13-s + 2.09·15-s + 0.299·17-s − 0.432·19-s + 2.97·21-s − 5.72·23-s − 0.590·25-s − 27-s − 0.718·29-s + 4.27·31-s − 0.138·33-s + 6.24·35-s − 3.92·37-s − 5.06·39-s + 7.57·41-s − 12.3·43-s − 2.09·45-s + 9.22·47-s + 1.84·49-s − 0.299·51-s + 1.26·53-s − 0.290·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.939·5-s − 1.12·7-s + 0.333·9-s + 0.0416·11-s + 1.40·13-s + 0.542·15-s + 0.0725·17-s − 0.0991·19-s + 0.649·21-s − 1.19·23-s − 0.118·25-s − 0.192·27-s − 0.133·29-s + 0.767·31-s − 0.0240·33-s + 1.05·35-s − 0.644·37-s − 0.810·39-s + 1.18·41-s − 1.88·43-s − 0.313·45-s + 1.34·47-s + 0.263·49-s − 0.0418·51-s + 0.173·53-s − 0.0391·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.09T + 5T^{2} \) |
| 7 | \( 1 + 2.97T + 7T^{2} \) |
| 11 | \( 1 - 0.138T + 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 - 0.299T + 17T^{2} \) |
| 19 | \( 1 + 0.432T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 + 0.718T + 29T^{2} \) |
| 31 | \( 1 - 4.27T + 31T^{2} \) |
| 37 | \( 1 + 3.92T + 37T^{2} \) |
| 41 | \( 1 - 7.57T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 9.22T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 + 3.01T + 59T^{2} \) |
| 61 | \( 1 - 2.98T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 7.54T + 97T^{2} \) |
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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
−7.45325489529972498270974659283, −6.63373219508609409925086038513, −6.21705726257624722230627124402, −5.55513578081057818827175562465, −4.55189049864615075364189038279, −3.70921900717218799495307710798, −3.50689841750590751728970377936, −2.23198996636842962619411684562, −0.970269029790897361051313473636, 0,
0.970269029790897361051313473636, 2.23198996636842962619411684562, 3.50689841750590751728970377936, 3.70921900717218799495307710798, 4.55189049864615075364189038279, 5.55513578081057818827175562465, 6.21705726257624722230627124402, 6.63373219508609409925086038513, 7.45325489529972498270974659283
