| L(s) = 1 | + 2-s + 2.73·3-s + 4-s + 0.477·5-s + 2.73·6-s − 4.17·7-s + 8-s + 4.47·9-s + 0.477·10-s − 3.05·11-s + 2.73·12-s − 0.721·13-s − 4.17·14-s + 1.30·15-s + 16-s + 1.18·17-s + 4.47·18-s − 19-s + 0.477·20-s − 11.4·21-s − 3.05·22-s − 6.92·23-s + 2.73·24-s − 4.77·25-s − 0.721·26-s + 4.03·27-s − 4.17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s + 0.213·5-s + 1.11·6-s − 1.57·7-s + 0.353·8-s + 1.49·9-s + 0.151·10-s − 0.921·11-s + 0.789·12-s − 0.200·13-s − 1.11·14-s + 0.337·15-s + 0.250·16-s + 0.287·17-s + 1.05·18-s − 0.229·19-s + 0.106·20-s − 2.49·21-s − 0.651·22-s − 1.44·23-s + 0.558·24-s − 0.954·25-s − 0.141·26-s + 0.775·27-s − 0.789·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 - 0.477T + 5T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 0.721T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 - 3.33T + 31T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 + 0.716T + 41T^{2} \) |
| 43 | \( 1 - 0.431T + 43T^{2} \) |
| 47 | \( 1 + 8.16T + 47T^{2} \) |
| 53 | \( 1 - 5.29T + 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 + 3.08T + 61T^{2} \) |
| 67 | \( 1 + 3.76T + 67T^{2} \) |
| 71 | \( 1 - 4.36T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 0.00212T + 79T^{2} \) |
| 83 | \( 1 - 7.69T + 83T^{2} \) |
| 89 | \( 1 + 5.27T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.58270901928199336160287913435, −6.84382018172514784904512570906, −6.05227037931845167314575281954, −5.50557424193986071513380792683, −4.34459816001136256593392981696, −3.67733889755591532619639375125, −3.17322510559133234047237638682, −2.46014755339585410298422684244, −1.85634414799992929634759643131, 0,
1.85634414799992929634759643131, 2.46014755339585410298422684244, 3.17322510559133234047237638682, 3.67733889755591532619639375125, 4.34459816001136256593392981696, 5.50557424193986071513380792683, 6.05227037931845167314575281954, 6.84382018172514784904512570906, 7.58270901928199336160287913435
