| L(s) = 1 | − 2.36·3-s + 2.61·5-s − 2.25·7-s + 2.61·9-s − 6.06·11-s + 5.42·13-s − 6.19·15-s − 17-s − 4.71·19-s + 5.34·21-s + 9.23·23-s + 1.82·25-s + 0.917·27-s + 4.85·29-s − 0.261·31-s + 14.3·33-s − 5.89·35-s + 4.91·37-s − 12.8·39-s − 5.42·41-s − 5.25·43-s + 6.82·45-s − 1.34·47-s − 1.91·49-s + 2.36·51-s + 5.39·53-s − 15.8·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 1.16·5-s − 0.852·7-s + 0.870·9-s − 1.82·11-s + 1.50·13-s − 1.59·15-s − 0.242·17-s − 1.08·19-s + 1.16·21-s + 1.92·23-s + 0.365·25-s + 0.176·27-s + 0.901·29-s − 0.0469·31-s + 2.49·33-s − 0.996·35-s + 0.807·37-s − 2.05·39-s − 0.848·41-s − 0.801·43-s + 1.01·45-s − 0.195·47-s − 0.273·49-s + 0.331·51-s + 0.740·53-s − 2.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.36T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 2.25T + 7T^{2} \) |
| 11 | \( 1 + 6.06T + 11T^{2} \) |
| 13 | \( 1 - 5.42T + 13T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 - 9.23T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + 0.261T + 31T^{2} \) |
| 37 | \( 1 - 4.91T + 37T^{2} \) |
| 41 | \( 1 + 5.42T + 41T^{2} \) |
| 43 | \( 1 + 5.25T + 43T^{2} \) |
| 47 | \( 1 + 1.34T + 47T^{2} \) |
| 53 | \( 1 - 5.39T + 53T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 3.44T + 67T^{2} \) |
| 71 | \( 1 + 2.07T + 71T^{2} \) |
| 73 | \( 1 + 0.551T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 7.01T + 83T^{2} \) |
| 89 | \( 1 + 8.76T + 89T^{2} \) |
| 97 | \( 1 + 1.65T + 97T^{2} \) |
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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
−8.162934404900803986723478935259, −6.95587378900937097733713860403, −6.37280630883751308291425553713, −5.95003682724910985638754628234, −5.23843154731192615724076653263, −4.66258928304263679301053232640, −3.27860752450531131492448594083, −2.46799208411170099191147324624, −1.20268058637710312574565973240, 0,
1.20268058637710312574565973240, 2.46799208411170099191147324624, 3.27860752450531131492448594083, 4.66258928304263679301053232640, 5.23843154731192615724076653263, 5.95003682724910985638754628234, 6.37280630883751308291425553713, 6.95587378900937097733713860403, 8.162934404900803986723478935259
