| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s − 6·13-s − 14-s + 16-s + 2·17-s − 3·18-s − 6·26-s − 28-s − 6·29-s + 8·31-s + 32-s + 2·34-s − 3·36-s + 10·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 6·52-s + 2·53-s − 56-s − 6·58-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 1.17·26-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.832·52-s + 0.274·53-s − 0.133·56-s − 0.787·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.82662078106542, −14.57257680348763, −13.98358591022275, −13.42310685691287, −12.92610959201452, −12.38592369923969, −11.90335956429882, −11.50641970733895, −10.96321971070093, −10.23399617411250, −9.722612589423903, −9.363351214934021, −8.539734659638513, −7.870634226968134, −7.548900111588559, −6.783225050277882, −6.252398580895870, −5.698956929482763, −5.089633438032180, −4.655159146843520, −3.850656645448166, −3.193624504926450, −2.575357287058195, −2.174647496205383, −0.9494134305738243, 0,
0.9494134305738243, 2.174647496205383, 2.575357287058195, 3.193624504926450, 3.850656645448166, 4.655159146843520, 5.089633438032180, 5.698956929482763, 6.252398580895870, 6.783225050277882, 7.548900111588559, 7.870634226968134, 8.539734659638513, 9.363351214934021, 9.722612589423903, 10.23399617411250, 10.96321971070093, 11.50641970733895, 11.90335956429882, 12.38592369923969, 12.92610959201452, 13.42310685691287, 13.98358591022275, 14.57257680348763, 14.82662078106542
