| L(s) = 1 | + 2.27·2-s − 3-s + 3.16·4-s − 3.11·5-s − 2.27·6-s − 7-s + 2.65·8-s + 9-s − 7.07·10-s + 3.06·11-s − 3.16·12-s + 1.48·13-s − 2.27·14-s + 3.11·15-s − 0.296·16-s + 2.04·17-s + 2.27·18-s − 3.13·19-s − 9.86·20-s + 21-s + 6.97·22-s + 2.08·23-s − 2.65·24-s + 4.69·25-s + 3.37·26-s − 27-s − 3.16·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s − 0.577·3-s + 1.58·4-s − 1.39·5-s − 0.928·6-s − 0.377·7-s + 0.939·8-s + 0.333·9-s − 2.23·10-s + 0.924·11-s − 0.914·12-s + 0.411·13-s − 0.607·14-s + 0.803·15-s − 0.0741·16-s + 0.494·17-s + 0.535·18-s − 0.720·19-s − 2.20·20-s + 0.218·21-s + 1.48·22-s + 0.433·23-s − 0.542·24-s + 0.938·25-s + 0.662·26-s − 0.192·27-s − 0.598·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 29 | \( 1 + 1.63T + 29T^{2} \) |
| 31 | \( 1 + 8.57T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 - 3.35T + 41T^{2} \) |
| 43 | \( 1 - 6.08T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 5.69T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 - 1.43T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 6.42T + 73T^{2} \) |
| 79 | \( 1 - 0.0149T + 79T^{2} \) |
| 83 | \( 1 - 7.86T + 83T^{2} \) |
| 89 | \( 1 + 8.03T + 89T^{2} \) |
| 97 | \( 1 + 7.38T + 97T^{2} \) |
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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
−7.34717440428036108017634154150, −6.52319565342261479051463147411, −6.01680752280412389325463389799, −5.36435078237549870413510240920, −4.33930217459178356602359737710, −4.12110233660537272696168634242, −3.48849467057024969499121369224, −2.67037181845850670303544134755, −1.33815013107279390548414481091, 0,
1.33815013107279390548414481091, 2.67037181845850670303544134755, 3.48849467057024969499121369224, 4.12110233660537272696168634242, 4.33930217459178356602359737710, 5.36435078237549870413510240920, 6.01680752280412389325463389799, 6.52319565342261479051463147411, 7.34717440428036108017634154150
