| L(s) = 1 | − 0.881·3-s + 2.46·5-s + 1.39·7-s − 2.22·9-s − 0.760·11-s − 3.31·13-s − 2.17·15-s + 2.14·17-s + 0.416·19-s − 1.23·21-s + 1.09·25-s + 4.60·27-s − 7.29·29-s − 7.39·31-s + 0.669·33-s + 3.44·35-s + 7.55·37-s + 2.91·39-s + 0.0737·41-s + 1.34·43-s − 5.48·45-s + 5.43·47-s − 5.04·49-s − 1.88·51-s − 7.32·53-s − 1.87·55-s − 0.367·57-s + ⋯ |
| L(s) = 1 | − 0.508·3-s + 1.10·5-s + 0.528·7-s − 0.740·9-s − 0.229·11-s − 0.918·13-s − 0.561·15-s + 0.519·17-s + 0.0955·19-s − 0.268·21-s + 0.218·25-s + 0.886·27-s − 1.35·29-s − 1.32·31-s + 0.116·33-s + 0.583·35-s + 1.24·37-s + 0.467·39-s + 0.0115·41-s + 0.204·43-s − 0.817·45-s + 0.793·47-s − 0.721·49-s − 0.264·51-s − 1.00·53-s − 0.252·55-s − 0.0486·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.881T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 11 | \( 1 + 0.760T + 11T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 - 0.416T + 19T^{2} \) |
| 29 | \( 1 + 7.29T + 29T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 - 7.55T + 37T^{2} \) |
| 41 | \( 1 - 0.0737T + 41T^{2} \) |
| 43 | \( 1 - 1.34T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 + 7.32T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 4.20T + 71T^{2} \) |
| 73 | \( 1 + 6.28T + 73T^{2} \) |
| 79 | \( 1 + 4.57T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 6.81T + 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−7.85271676301761034247046461803, −7.43051885280097440426214807665, −6.31469292752256436629081775593, −5.74243309592567614079088960305, −5.26297691086342743763530419680, −4.48868481305183092825837910781, −3.23230938615260306920248855120, −2.34540412074306358990396610274, −1.50348501760894694313632997160, 0,
1.50348501760894694313632997160, 2.34540412074306358990396610274, 3.23230938615260306920248855120, 4.48868481305183092825837910781, 5.26297691086342743763530419680, 5.74243309592567614079088960305, 6.31469292752256436629081775593, 7.43051885280097440426214807665, 7.85271676301761034247046461803
