L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 2.76·5-s + 6·6-s + 12.7·7-s − 8·8-s + 9·9-s + 5.52·10-s + 57.3·11-s − 12·12-s + 25.0·13-s − 25.5·14-s + 8.28·15-s + 16·16-s − 20.2·17-s − 18·18-s − 11.0·20-s − 38.2·21-s − 114.·22-s − 35.0·23-s + 24·24-s − 117.·25-s − 50.0·26-s − 27·27-s + 51.0·28-s + 76.0·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.246·5-s + 0.408·6-s + 0.689·7-s − 0.353·8-s + 0.333·9-s + 0.174·10-s + 1.57·11-s − 0.288·12-s + 0.534·13-s − 0.487·14-s + 0.142·15-s + 0.250·16-s − 0.289·17-s − 0.235·18-s − 0.123·20-s − 0.397·21-s − 1.11·22-s − 0.317·23-s + 0.204·24-s − 0.938·25-s − 0.377·26-s − 0.192·27-s + 0.344·28-s + 0.487·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 2.76T + 125T^{2} \) |
| 7 | \( 1 - 12.7T + 343T^{2} \) |
| 11 | \( 1 - 57.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 25.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 35.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 76.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 95.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 285.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 436.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 292.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 128.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 510.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 234.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 6.63T + 3.00e5T^{2} \) |
| 71 | \( 1 - 176.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 267.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 921.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 952.T + 9.12e5T^{2} \) |
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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
−8.310173037267654186026798821109, −7.70397435702047172528318235028, −6.62629619260342946874191236434, −6.30674290378823228230212649082, −5.18110450558819193356734512492, −4.24332751412047708094808806583, −3.42830089885653623168102430679, −1.86929041554479355546611669413, −1.22753397428462921275193306004, 0,
1.22753397428462921275193306004, 1.86929041554479355546611669413, 3.42830089885653623168102430679, 4.24332751412047708094808806583, 5.18110450558819193356734512492, 6.30674290378823228230212649082, 6.62629619260342946874191236434, 7.70397435702047172528318235028, 8.310173037267654186026798821109