| L(s) = 1 | − 3.37·5-s − 4.70·7-s − 0.875·11-s + 1.37·13-s + 2.37·17-s + 5.57·19-s + 4.70·23-s + 6.37·25-s − 5.37·29-s + 6.45·31-s + 15.8·35-s − 4·37-s − 41-s − 0.875·43-s + 4.70·47-s + 15.1·49-s − 4·53-s + 2.95·55-s − 8.53·59-s + 2.11·61-s − 4.62·65-s − 8.53·67-s + 9.40·71-s + 10.3·73-s + 4.11·77-s − 6.45·79-s + 2.95·83-s + ⋯ |
| L(s) = 1 | − 1.50·5-s − 1.77·7-s − 0.263·11-s + 0.380·13-s + 0.575·17-s + 1.27·19-s + 0.980·23-s + 1.27·25-s − 0.997·29-s + 1.15·31-s + 2.68·35-s − 0.657·37-s − 0.156·41-s − 0.133·43-s + 0.685·47-s + 2.15·49-s − 0.549·53-s + 0.398·55-s − 1.11·59-s + 0.271·61-s − 0.573·65-s − 1.04·67-s + 1.11·71-s + 1.21·73-s + 0.469·77-s − 0.726·79-s + 0.324·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 11 | \( 1 + 0.875T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 - 6.45T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 + 0.875T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 8.53T + 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 + 8.53T + 67T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 6.45T + 79T^{2} \) |
| 83 | \( 1 - 2.95T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−7.70667940185301408469364328192, −7.22550908637892626595478931064, −6.55778579436150874946960669554, −5.72732758305758865164229543020, −4.87253432852668878120270537488, −3.83285657836590094750976037086, −3.36805096861447066646272745118, −2.80576497505157332918829586907, −1.00757250932946861833644876816, 0,
1.00757250932946861833644876816, 2.80576497505157332918829586907, 3.36805096861447066646272745118, 3.83285657836590094750976037086, 4.87253432852668878120270537488, 5.72732758305758865164229543020, 6.55778579436150874946960669554, 7.22550908637892626595478931064, 7.70667940185301408469364328192
