| L(s) = 1 | + 2-s + 1.34·3-s + 4-s − 5-s + 1.34·6-s + 2.38·7-s + 8-s − 1.19·9-s − 10-s − 3.62·11-s + 1.34·12-s − 2.92·13-s + 2.38·14-s − 1.34·15-s + 16-s − 2.81·17-s − 1.19·18-s + 1.26·19-s − 20-s + 3.19·21-s − 3.62·22-s − 5.29·23-s + 1.34·24-s + 25-s − 2.92·26-s − 5.63·27-s + 2.38·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.774·3-s + 0.5·4-s − 0.447·5-s + 0.547·6-s + 0.900·7-s + 0.353·8-s − 0.399·9-s − 0.316·10-s − 1.09·11-s + 0.387·12-s − 0.810·13-s + 0.636·14-s − 0.346·15-s + 0.250·16-s − 0.682·17-s − 0.282·18-s + 0.291·19-s − 0.223·20-s + 0.697·21-s − 0.771·22-s − 1.10·23-s + 0.273·24-s + 0.200·25-s − 0.573·26-s − 1.08·27-s + 0.450·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 1.21T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 + 4.43T + 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 + 7.47T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 + 6.18T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 6.82T + 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 4.26T + 79T^{2} \) |
| 83 | \( 1 + 5.16T + 83T^{2} \) |
| 89 | \( 1 + 0.599T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−7.74492269376399358048543521002, −7.21829801839160252207729252745, −6.22186392245860094867307304960, −5.36391949879822350586077782569, −4.78235912149010041122138737209, −4.11387179796411397555843779119, −3.11463329000768207585161597599, −2.53935553381052187095626557774, −1.74839922206517247373086384307, 0,
1.74839922206517247373086384307, 2.53935553381052187095626557774, 3.11463329000768207585161597599, 4.11387179796411397555843779119, 4.78235912149010041122138737209, 5.36391949879822350586077782569, 6.22186392245860094867307304960, 7.21829801839160252207729252745, 7.74492269376399358048543521002
