| L(s) = 1 | + 2·2-s − 2.47·3-s + 4·4-s − 14.9·5-s − 4.95·6-s + 7·7-s + 8·8-s − 20.8·9-s − 29.9·10-s + 15.6·11-s − 9.90·12-s + 14·14-s + 37.0·15-s + 16·16-s + 105.·17-s − 41.7·18-s − 145.·19-s − 59.8·20-s − 17.3·21-s + 31.3·22-s + 146.·23-s − 19.8·24-s + 98.9·25-s + 118.·27-s + 28·28-s − 278.·29-s + 74.1·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.476·3-s + 0.5·4-s − 1.33·5-s − 0.337·6-s + 0.377·7-s + 0.353·8-s − 0.772·9-s − 0.946·10-s + 0.430·11-s − 0.238·12-s + 0.267·14-s + 0.637·15-s + 0.250·16-s + 1.50·17-s − 0.546·18-s − 1.75·19-s − 0.669·20-s − 0.180·21-s + 0.304·22-s + 1.32·23-s − 0.168·24-s + 0.791·25-s + 0.844·27-s + 0.188·28-s − 1.78·29-s + 0.451·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.47T + 27T^{2} \) |
| 5 | \( 1 + 14.9T + 125T^{2} \) |
| 11 | \( 1 - 15.6T + 1.33e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 145.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 146.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 237.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 229.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 40.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 364.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 191.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 181.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 654.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 420.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 663.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 49.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 635.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 49.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.78e3T + 9.12e5T^{2} \) |
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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
−8.215656096918459156249878560289, −7.37119444233865318894172420266, −6.66772554063026622994494245114, −5.73353221452164705798873707841, −5.07102239287988055251601830546, −4.14909401570216188273287583088, −3.55406023038711281996351644617, −2.55638362993802964983275883527, −1.12680430924178666094422842262, 0,
1.12680430924178666094422842262, 2.55638362993802964983275883527, 3.55406023038711281996351644617, 4.14909401570216188273287583088, 5.07102239287988055251601830546, 5.73353221452164705798873707841, 6.66772554063026622994494245114, 7.37119444233865318894172420266, 8.215656096918459156249878560289
