L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·13-s − 14-s + 16-s + 6·17-s − 4·19-s − 23-s − 2·26-s − 28-s − 6·29-s − 4·31-s + 32-s + 6·34-s − 2·37-s − 4·38-s + 6·41-s − 8·43-s − 46-s + 12·47-s + 49-s − 2·52-s − 6·53-s − 56-s − 6·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.208·23-s − 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s − 0.648·38-s + 0.937·41-s − 1.21·43-s − 0.147·46-s + 1.75·47-s + 1/7·49-s − 0.277·52-s − 0.824·53-s − 0.133·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.45505596691552, −13.87261677947151, −13.25144952142663, −12.91056377612737, −12.35984149050036, −12.02239329290398, −11.50624386460036, −10.71783101193543, −10.54381012043473, −9.777967111809772, −9.446831514174184, −8.744945739305967, −8.126769316493535, −7.487739921455522, −7.214474763558102, −6.481395570013934, −5.923333862041864, −5.470181369260840, −4.959720877379336, −4.193558580000975, −3.693916990337529, −3.200292801849616, −2.406490460158183, −1.899584507198845, −0.9826577168300783, 0,
0.9826577168300783, 1.899584507198845, 2.406490460158183, 3.200292801849616, 3.693916990337529, 4.193558580000975, 4.959720877379336, 5.470181369260840, 5.923333862041864, 6.481395570013934, 7.214474763558102, 7.487739921455522, 8.126769316493535, 8.744945739305967, 9.446831514174184, 9.777967111809772, 10.54381012043473, 10.71783101193543, 11.50624386460036, 12.02239329290398, 12.35984149050036, 12.91056377612737, 13.25144952142663, 13.87261677947151, 14.45505596691552