L(s) = 1 | + 2-s − 3-s + 4-s − 0.646·5-s − 6-s + 4.78·7-s + 8-s + 9-s − 0.646·10-s − 2.66·11-s − 12-s − 0.293·13-s + 4.78·14-s + 0.646·15-s + 16-s − 17-s + 18-s − 7.66·19-s − 0.646·20-s − 4.78·21-s − 2.66·22-s + 3.48·23-s − 24-s − 4.58·25-s − 0.293·26-s − 27-s + 4.78·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.289·5-s − 0.408·6-s + 1.80·7-s + 0.353·8-s + 0.333·9-s − 0.204·10-s − 0.803·11-s − 0.288·12-s − 0.0815·13-s + 1.27·14-s + 0.166·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 1.75·19-s − 0.144·20-s − 1.04·21-s − 0.568·22-s + 0.727·23-s − 0.204·24-s − 0.916·25-s − 0.0576·26-s − 0.192·27-s + 0.904·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 0.646T + 5T^{2} \) |
| 7 | \( 1 - 4.78T + 7T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 + 0.293T + 13T^{2} \) |
| 19 | \( 1 + 7.66T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 - 0.738T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 + 6.52T + 43T^{2} \) |
| 47 | \( 1 + 4.77T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 1.01T + 71T^{2} \) |
| 73 | \( 1 + 6.75T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 7.51T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.48201772116401143862057528254, −7.16784560640767073681466495340, −6.02911381848725775810635452733, −5.51500868474334174498316515963, −4.63699858516921735195057270682, −4.48407264617938338355902870558, −3.39353826793180773639116072574, −2.14109722203793144200454221721, −1.63031001963286599863906523773, 0,
1.63031001963286599863906523773, 2.14109722203793144200454221721, 3.39353826793180773639116072574, 4.48407264617938338355902870558, 4.63699858516921735195057270682, 5.51500868474334174498316515963, 6.02911381848725775810635452733, 7.16784560640767073681466495340, 7.48201772116401143862057528254