| L(s) = 1 | − 0.567·3-s − 2.51·5-s − 3.57·7-s − 2.67·9-s + 5.90·11-s − 3.72·13-s + 1.42·15-s + 5.80·17-s + 1.27·19-s + 2.02·21-s + 3.04·23-s + 1.31·25-s + 3.22·27-s − 3.75·29-s + 7.64·31-s − 3.34·33-s + 8.97·35-s + 11.9·37-s + 2.11·39-s − 9.65·41-s + 1.48·43-s + 6.73·45-s − 0.516·47-s + 5.74·49-s − 3.29·51-s − 10.5·53-s − 14.8·55-s + ⋯ |
| L(s) = 1 | − 0.327·3-s − 1.12·5-s − 1.34·7-s − 0.892·9-s + 1.77·11-s − 1.03·13-s + 0.368·15-s + 1.40·17-s + 0.292·19-s + 0.442·21-s + 0.634·23-s + 0.263·25-s + 0.620·27-s − 0.696·29-s + 1.37·31-s − 0.582·33-s + 1.51·35-s + 1.96·37-s + 0.338·39-s − 1.50·41-s + 0.227·43-s + 1.00·45-s − 0.0753·47-s + 0.821·49-s − 0.461·51-s − 1.45·53-s − 1.99·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 0.567T + 3T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 - 7.64T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 - 1.48T + 43T^{2} \) |
| 47 | \( 1 + 0.516T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 0.696T + 59T^{2} \) |
| 61 | \( 1 + 4.04T + 61T^{2} \) |
| 67 | \( 1 - 8.27T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 0.498T + 73T^{2} \) |
| 79 | \( 1 + 3.99T + 79T^{2} \) |
| 83 | \( 1 + 2.57T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.987511865481177308876873989586, −7.33875326852956968472616296515, −6.52780185534445421506506758564, −6.06824550737089675024171432982, −5.05874667204666190449182029735, −4.13376212957735884189039201397, −3.37616019192193965195247727121, −2.84638889492071245135450619967, −1.08360940815577404027161782992, 0,
1.08360940815577404027161782992, 2.84638889492071245135450619967, 3.37616019192193965195247727121, 4.13376212957735884189039201397, 5.05874667204666190449182029735, 6.06824550737089675024171432982, 6.52780185534445421506506758564, 7.33875326852956968472616296515, 7.987511865481177308876873989586
