| L(s) = 1 | + 3.06·3-s + 5-s − 4.41·7-s + 6.40·9-s − 4.27·11-s − 2.79·13-s + 3.06·15-s − 4.43·17-s − 2.43·19-s − 13.5·21-s − 5.77·23-s + 25-s + 10.4·27-s + 0.409·29-s − 7.79·31-s − 13.1·33-s − 4.41·35-s − 37-s − 8.58·39-s + 0.757·41-s − 2.19·43-s + 6.40·45-s + 4.26·47-s + 12.4·49-s − 13.6·51-s − 0.137·53-s − 4.27·55-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 0.447·5-s − 1.66·7-s + 2.13·9-s − 1.28·11-s − 0.776·13-s + 0.791·15-s − 1.07·17-s − 0.558·19-s − 2.95·21-s − 1.20·23-s + 0.200·25-s + 2.01·27-s + 0.0759·29-s − 1.39·31-s − 2.28·33-s − 0.746·35-s − 0.164·37-s − 1.37·39-s + 0.118·41-s − 0.335·43-s + 0.955·45-s + 0.622·47-s + 1.78·49-s − 1.90·51-s − 0.0188·53-s − 0.575·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 3.06T + 3T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 + 5.77T + 23T^{2} \) |
| 29 | \( 1 - 0.409T + 29T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 41 | \( 1 - 0.757T + 41T^{2} \) |
| 43 | \( 1 + 2.19T + 43T^{2} \) |
| 47 | \( 1 - 4.26T + 47T^{2} \) |
| 53 | \( 1 + 0.137T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 + 3.02T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.20T + 73T^{2} \) |
| 79 | \( 1 + 7.11T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.479219591567673774439297140542, −7.69165119299709671849118052069, −7.01607043776188563827600889284, −6.30017727878481408787317735887, −5.22846951701874374382315373650, −4.09413750010268114460292356372, −3.40150999301035086235205884531, −2.45848808150595997945896068719, −2.18701206860746762720220146214, 0,
2.18701206860746762720220146214, 2.45848808150595997945896068719, 3.40150999301035086235205884531, 4.09413750010268114460292356372, 5.22846951701874374382315373650, 6.30017727878481408787317735887, 7.01607043776188563827600889284, 7.69165119299709671849118052069, 8.479219591567673774439297140542
