| L(s) = 1 | + 2-s − 2.82·3-s + 4-s + 0.779·5-s − 2.82·6-s − 3.19·7-s + 8-s + 4.96·9-s + 0.779·10-s − 0.388·11-s − 2.82·12-s + 1.07·13-s − 3.19·14-s − 2.20·15-s + 16-s − 5.37·17-s + 4.96·18-s + 6.70·19-s + 0.779·20-s + 9.00·21-s − 0.388·22-s + 5.06·23-s − 2.82·24-s − 4.39·25-s + 1.07·26-s − 5.55·27-s − 3.19·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.62·3-s + 0.5·4-s + 0.348·5-s − 1.15·6-s − 1.20·7-s + 0.353·8-s + 1.65·9-s + 0.246·10-s − 0.117·11-s − 0.814·12-s + 0.298·13-s − 0.852·14-s − 0.568·15-s + 0.250·16-s − 1.30·17-s + 1.17·18-s + 1.53·19-s + 0.174·20-s + 1.96·21-s − 0.0828·22-s + 1.05·23-s − 0.576·24-s − 0.878·25-s + 0.210·26-s − 1.06·27-s − 0.603·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.779T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 + 0.388T + 11T^{2} \) |
| 13 | \( 1 - 1.07T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 4.46T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 - 9.75T + 53T^{2} \) |
| 59 | \( 1 + 1.74T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 9.88T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 + 5.00T + 73T^{2} \) |
| 79 | \( 1 + 2.25T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 + 9.36T + 97T^{2} \) |
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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
−7.66481048274662813542938908832, −6.93293714436293787660908073144, −6.45308467162506073767535385253, −5.75103313107039230996477552715, −5.32636280931029404541232800100, −4.43916974892500859114146597739, −3.58904757535881273762524304658, −2.59671676960319859632679166953, −1.25662998424907381108041927779, 0,
1.25662998424907381108041927779, 2.59671676960319859632679166953, 3.58904757535881273762524304658, 4.43916974892500859114146597739, 5.32636280931029404541232800100, 5.75103313107039230996477552715, 6.45308467162506073767535385253, 6.93293714436293787660908073144, 7.66481048274662813542938908832
