| L(s) = 1 | + 0.492·3-s + 5-s + 7-s − 2.75·9-s + 6.19·11-s − 13-s + 0.492·15-s − 1.82·17-s + 3.32·19-s + 0.492·21-s − 3.20·23-s + 25-s − 2.83·27-s + 4.71·29-s + 2.49·31-s + 3.04·33-s + 35-s − 3.63·37-s − 0.492·39-s + 2.77·41-s + 10.7·43-s − 2.75·45-s + 10.3·47-s + 49-s − 0.895·51-s − 3.40·53-s + 6.19·55-s + ⋯ |
| L(s) = 1 | + 0.284·3-s + 0.447·5-s + 0.377·7-s − 0.919·9-s + 1.86·11-s − 0.277·13-s + 0.127·15-s − 0.441·17-s + 0.762·19-s + 0.107·21-s − 0.669·23-s + 0.200·25-s − 0.545·27-s + 0.875·29-s + 0.447·31-s + 0.530·33-s + 0.169·35-s − 0.598·37-s − 0.0787·39-s + 0.433·41-s + 1.64·43-s − 0.411·45-s + 1.50·47-s + 0.142·49-s − 0.125·51-s − 0.467·53-s + 0.835·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.529340750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.529340750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.492T + 3T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 - 4.71T + 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 + 3.63T + 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 - 2.72T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.705370892563897391348280768953, −7.86734858776152168118484030086, −7.03814150001452676178355481502, −6.23314002681568643875686558632, −5.70299934539581270559273021033, −4.65203210254914192477430034950, −3.91269948271840029771309105228, −2.94239659047831780619080185458, −2.02781678801375584170557433481, −0.959423927009935577706155451707,
0.959423927009935577706155451707, 2.02781678801375584170557433481, 2.94239659047831780619080185458, 3.91269948271840029771309105228, 4.65203210254914192477430034950, 5.70299934539581270559273021033, 6.23314002681568643875686558632, 7.03814150001452676178355481502, 7.86734858776152168118484030086, 8.705370892563897391348280768953
