| L(s) = 1 | − 0.279·3-s − 0.263·5-s − 1.45·7-s − 2.92·9-s + 0.639·11-s − 3.16·13-s + 0.0735·15-s + 17-s − 2.43·19-s + 0.407·21-s − 4.24·23-s − 4.93·25-s + 1.65·27-s + 0.285·29-s − 1.45·31-s − 0.178·33-s + 0.384·35-s − 5.17·37-s + 0.885·39-s + 9.84·41-s − 7.20·43-s + 0.769·45-s + 0.0570·47-s − 4.87·49-s − 0.279·51-s + 4.71·53-s − 0.168·55-s + ⋯ |
| L(s) = 1 | − 0.161·3-s − 0.117·5-s − 0.551·7-s − 0.974·9-s + 0.192·11-s − 0.879·13-s + 0.0189·15-s + 0.242·17-s − 0.559·19-s + 0.0888·21-s − 0.884·23-s − 0.986·25-s + 0.318·27-s + 0.0530·29-s − 0.260·31-s − 0.0311·33-s + 0.0649·35-s − 0.851·37-s + 0.141·39-s + 1.53·41-s − 1.09·43-s + 0.114·45-s + 0.00832·47-s − 0.696·49-s − 0.0391·51-s + 0.647·53-s − 0.0227·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7454214200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7454214200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 0.279T + 3T^{2} \) |
| 5 | \( 1 + 0.263T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 - 0.639T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 0.285T + 29T^{2} \) |
| 31 | \( 1 + 1.45T + 31T^{2} \) |
| 37 | \( 1 + 5.17T + 37T^{2} \) |
| 41 | \( 1 - 9.84T + 41T^{2} \) |
| 43 | \( 1 + 7.20T + 43T^{2} \) |
| 47 | \( 1 - 0.0570T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 61 | \( 1 - 0.366T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 + 4.85T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 2.05T + 83T^{2} \) |
| 89 | \( 1 - 7.90T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.975699000434884171493437715146, −7.05734820357539389180764746010, −6.43812566103167121899496525686, −5.75028013257457414965598342113, −5.17305729420517218035383923110, −4.21576426306986872086846968582, −3.52144257255541671464011654205, −2.66738023802846528567671278799, −1.89883154070519858073724221152, −0.40621813450551173845995600021,
0.40621813450551173845995600021, 1.89883154070519858073724221152, 2.66738023802846528567671278799, 3.52144257255541671464011654205, 4.21576426306986872086846968582, 5.17305729420517218035383923110, 5.75028013257457414965598342113, 6.43812566103167121899496525686, 7.05734820357539389180764746010, 7.975699000434884171493437715146
