L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 3·11-s − 13-s + 15-s − 17-s + 7·19-s + 2·21-s + 4·23-s + 25-s − 27-s + 4·29-s − 11·31-s − 3·33-s + 2·35-s + 9·37-s + 39-s + 4·41-s + 11·43-s − 45-s − 6·47-s − 3·49-s + 51-s − 6·53-s − 3·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 1.60·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.97·31-s − 0.522·33-s + 0.338·35-s + 1.47·37-s + 0.160·39-s + 0.624·41-s + 1.67·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.97652082929700, −12.77803068532757, −12.30894953807878, −11.84936598676423, −11.32034056452673, −11.02028852225198, −10.62371371019725, −9.699593218756944, −9.560591443800734, −9.201863920047116, −8.628793860475056, −7.769396431584436, −7.522269121688411, −7.035004541220423, −6.495840811317039, −6.060554691848634, −5.560686252389375, −4.931645235544507, −4.470833244708882, −3.900022720077725, −3.272236250497722, −2.964664007535975, −2.098003018238758, −1.255374750478727, −0.8091701366577640, 0,
0.8091701366577640, 1.255374750478727, 2.098003018238758, 2.964664007535975, 3.272236250497722, 3.900022720077725, 4.470833244708882, 4.931645235544507, 5.560686252389375, 6.060554691848634, 6.495840811317039, 7.035004541220423, 7.522269121688411, 7.769396431584436, 8.628793860475056, 9.201863920047116, 9.560591443800734, 9.699593218756944, 10.62371371019725, 11.02028852225198, 11.32034056452673, 11.84936598676423, 12.30894953807878, 12.77803068532757, 12.97652082929700