L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s − 3·5-s + 6·6-s − 7-s + 6·9-s + 6·10-s + 11-s − 6·12-s − 13-s + 2·14-s + 9·15-s − 4·16-s − 8·17-s − 12·18-s − 4·19-s − 6·20-s + 3·21-s − 2·22-s − 9·23-s + 4·25-s + 2·26-s − 9·27-s − 2·28-s − 8·29-s − 18·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s − 1.34·5-s + 2.44·6-s − 0.377·7-s + 2·9-s + 1.89·10-s + 0.301·11-s − 1.73·12-s − 0.277·13-s + 0.534·14-s + 2.32·15-s − 16-s − 1.94·17-s − 2.82·18-s − 0.917·19-s − 1.34·20-s + 0.654·21-s − 0.426·22-s − 1.87·23-s + 4/5·25-s + 0.392·26-s − 1.73·27-s − 0.377·28-s − 1.48·29-s − 3.28·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−9.177678656566865799281343761624, −8.329663666284703734497099317122, −7.42352143997392678851090289609, −6.77488664651047884179804084466, −6.03685525577675971166905423928, −4.59040983008654711676994142040, −4.05990015475203683261300091326, −1.85721237479120013699100313692, 0, 0,
1.85721237479120013699100313692, 4.05990015475203683261300091326, 4.59040983008654711676994142040, 6.03685525577675971166905423928, 6.77488664651047884179804084466, 7.42352143997392678851090289609, 8.329663666284703734497099317122, 9.177678656566865799281343761624