L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 2·13-s + 16-s − 6·17-s − 8·19-s + 20-s + 25-s − 2·26-s − 6·29-s + 4·31-s + 32-s − 6·34-s − 10·37-s − 8·38-s + 40-s − 6·41-s − 4·43-s + 50-s − 2·52-s + 6·53-s − 6·58-s − 12·59-s + 10·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 1.83·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 1.64·37-s − 1.29·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s + 0.141·50-s − 0.277·52-s + 0.824·53-s − 0.787·58-s − 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.988751310608497948309425656861, −6.88089320758720461322150711491, −6.63010526050831562050636982417, −5.75531801300200717416192667829, −4.92567559830316004423556337525, −4.33770231717830777250549003341, −3.45283173381757043345013171615, −2.33971670719236581491979893835, −1.84101162653270326557570577442, 0,
1.84101162653270326557570577442, 2.33971670719236581491979893835, 3.45283173381757043345013171615, 4.33770231717830777250549003341, 4.92567559830316004423556337525, 5.75531801300200717416192667829, 6.63010526050831562050636982417, 6.88089320758720461322150711491, 7.988751310608497948309425656861