| L(s) = 1 | + 2-s − 0.219·3-s + 4-s − 0.219·6-s + 1.81·7-s + 8-s − 2.95·9-s + 4.42·11-s − 0.219·12-s + 1.81·14-s + 16-s + 6.15·17-s − 2.95·18-s − 6.70·19-s − 0.399·21-s + 4.42·22-s − 2.02·23-s − 0.219·24-s + 1.30·27-s + 1.81·28-s + 3.67·29-s + 7.90·31-s + 32-s − 0.971·33-s + 6.15·34-s − 2.95·36-s + 0.380·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.126·3-s + 0.5·4-s − 0.0896·6-s + 0.687·7-s + 0.353·8-s − 0.983·9-s + 1.33·11-s − 0.0634·12-s + 0.486·14-s + 0.250·16-s + 1.49·17-s − 0.695·18-s − 1.53·19-s − 0.0872·21-s + 0.942·22-s − 0.422·23-s − 0.0448·24-s + 0.251·27-s + 0.343·28-s + 0.682·29-s + 1.41·31-s + 0.176·32-s − 0.169·33-s + 1.05·34-s − 0.491·36-s + 0.0625·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.684628086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.684628086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.219T + 3T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 17 | \( 1 - 6.15T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 - 3.67T + 29T^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 - 0.380T + 37T^{2} \) |
| 41 | \( 1 + 7.27T + 41T^{2} \) |
| 43 | \( 1 + 0.648T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 - 7.03T + 53T^{2} \) |
| 59 | \( 1 - 5.42T + 59T^{2} \) |
| 61 | \( 1 - 6.01T + 61T^{2} \) |
| 67 | \( 1 + 4.00T + 67T^{2} \) |
| 71 | \( 1 - 1.31T + 71T^{2} \) |
| 73 | \( 1 + 8.07T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + 0.575T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.77793489286747363411301217767, −6.91845778532570061392141597027, −6.22969340328634627974902647554, −5.79857585119497561361257669908, −4.95140667308099652893522631714, −4.29543256649665444173591467166, −3.59530749205367561538667564849, −2.75715740550442499448302484461, −1.85033735885566829526689472822, −0.879423690162416218872493485239,
0.879423690162416218872493485239, 1.85033735885566829526689472822, 2.75715740550442499448302484461, 3.59530749205367561538667564849, 4.29543256649665444173591467166, 4.95140667308099652893522631714, 5.79857585119497561361257669908, 6.22969340328634627974902647554, 6.91845778532570061392141597027, 7.77793489286747363411301217767
