L(s) = 1 | − 2.51·2-s − 3-s + 4.32·4-s + 2.51·6-s + 0.514·7-s − 5.83·8-s + 9-s − 0.806·11-s − 4.32·12-s − 13-s − 1.29·14-s + 6.02·16-s + 2.02·17-s − 2.51·18-s + 0.292·19-s − 0.514·21-s + 2.02·22-s − 8.34·23-s + 5.83·24-s + 2.51·26-s − 27-s + 2.22·28-s − 9.64·29-s + 6.22·31-s − 3.48·32-s + 0.806·33-s − 5.09·34-s + ⋯ |
L(s) = 1 | − 1.77·2-s − 0.577·3-s + 2.16·4-s + 1.02·6-s + 0.194·7-s − 2.06·8-s + 0.333·9-s − 0.243·11-s − 1.24·12-s − 0.277·13-s − 0.345·14-s + 1.50·16-s + 0.491·17-s − 0.592·18-s + 0.0671·19-s − 0.112·21-s + 0.432·22-s − 1.74·23-s + 1.19·24-s + 0.493·26-s − 0.192·27-s + 0.419·28-s − 1.79·29-s + 1.11·31-s − 0.616·32-s + 0.140·33-s − 0.874·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 7 | \( 1 - 0.514T + 7T^{2} \) |
| 11 | \( 1 + 0.806T + 11T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 - 0.292T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 + 9.64T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 - 6.34T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 - 7.96T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 4.70T + 53T^{2} \) |
| 59 | \( 1 - 4.12T + 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 - 6.57T + 83T^{2} \) |
| 89 | \( 1 + 0.971T + 89T^{2} \) |
| 97 | \( 1 + 9.61T + 97T^{2} \) |
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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
−9.673184490035068719851929271002, −8.866362479062510072994527433662, −7.78007775450077151448623043303, −7.54711719853155179812786791618, −6.34201168444215658603698384617, −5.64541376039925194511285020555, −4.20288599748644012999333815357, −2.58615382366371990052800286217, −1.43048443842338719105823122216, 0,
1.43048443842338719105823122216, 2.58615382366371990052800286217, 4.20288599748644012999333815357, 5.64541376039925194511285020555, 6.34201168444215658603698384617, 7.54711719853155179812786791618, 7.78007775450077151448623043303, 8.866362479062510072994527433662, 9.673184490035068719851929271002