| L(s) = 1 | + 2-s + 3.36·3-s + 4-s − 5-s + 3.36·6-s + 3.49·7-s + 8-s + 8.32·9-s − 10-s − 1.63·11-s + 3.36·12-s + 3.62·13-s + 3.49·14-s − 3.36·15-s + 16-s + 1.12·17-s + 8.32·18-s − 5.25·19-s − 20-s + 11.7·21-s − 1.63·22-s + 4.26·23-s + 3.36·24-s + 25-s + 3.62·26-s + 17.9·27-s + 3.49·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.94·3-s + 0.5·4-s − 0.447·5-s + 1.37·6-s + 1.32·7-s + 0.353·8-s + 2.77·9-s − 0.316·10-s − 0.491·11-s + 0.971·12-s + 1.00·13-s + 0.933·14-s − 0.869·15-s + 0.250·16-s + 0.273·17-s + 1.96·18-s − 1.20·19-s − 0.223·20-s + 2.56·21-s − 0.347·22-s + 0.888·23-s + 0.687·24-s + 0.200·25-s + 0.711·26-s + 3.45·27-s + 0.660·28-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
| \(L(1)\) |
\(\approx\) |
\(7.627581329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.627581329\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.36T + 3T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 - 3.62T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 0.741T + 31T^{2} \) |
| 37 | \( 1 + 8.18T + 37T^{2} \) |
| 41 | \( 1 + 4.04T + 41T^{2} \) |
| 43 | \( 1 - 3.68T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 - 7.90T + 59T^{2} \) |
| 61 | \( 1 + 8.88T + 61T^{2} \) |
| 67 | \( 1 + 1.05T + 67T^{2} \) |
| 71 | \( 1 + 0.284T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 1.81T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.120692513856853217672656360765, −7.41014230670836515940120763995, −7.08005536256700919253812247644, −5.82179965662146744268318638150, −4.91067702764430911110008498089, −4.17716619342921414826696746909, −3.70171676305903344349777610289, −2.89648977032295553076521513268, −2.01801927315980810885969021266, −1.41968602258175421542279515030,
1.41968602258175421542279515030, 2.01801927315980810885969021266, 2.89648977032295553076521513268, 3.70171676305903344349777610289, 4.17716619342921414826696746909, 4.91067702764430911110008498089, 5.82179965662146744268318638150, 7.08005536256700919253812247644, 7.41014230670836515940120763995, 8.120692513856853217672656360765
