| L(s) = 1 | + 3-s − 2·7-s − 2·9-s + 11-s + 4·13-s + 2·17-s − 2·21-s − 23-s − 5·27-s − 7·31-s + 33-s + 3·37-s + 4·39-s − 8·41-s + 6·43-s + 8·47-s − 3·49-s + 2·51-s − 6·53-s + 5·59-s − 12·61-s + 4·63-s + 7·67-s − 69-s + 3·71-s − 4·73-s − 2·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.301·11-s + 1.10·13-s + 0.485·17-s − 0.436·21-s − 0.208·23-s − 0.962·27-s − 1.25·31-s + 0.174·33-s + 0.493·37-s + 0.640·39-s − 1.24·41-s + 0.914·43-s + 1.16·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s + 0.650·59-s − 1.53·61-s + 0.503·63-s + 0.855·67-s − 0.120·69-s + 0.356·71-s − 0.468·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| 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
−16.04785874632436, −15.65673704318561, −14.79096888006822, −14.60828201917795, −13.77135387009324, −13.52553773024894, −12.90457119397648, −12.21882159476438, −11.77597882113796, −10.85675135986489, −10.75418666876266, −9.695297507041682, −9.314040694444650, −8.829261825871737, −8.147321907860788, −7.682489099139665, −6.846527463665715, −6.219142877398871, −5.750778726790997, −5.022343015831035, −3.910991242501951, −3.581130191778624, −2.916010512263472, −2.099858742760242, −1.156325624990347, 0,
1.156325624990347, 2.099858742760242, 2.916010512263472, 3.581130191778624, 3.910991242501951, 5.022343015831035, 5.750778726790997, 6.219142877398871, 6.846527463665715, 7.682489099139665, 8.147321907860788, 8.829261825871737, 9.314040694444650, 9.695297507041682, 10.75418666876266, 10.85675135986489, 11.77597882113796, 12.21882159476438, 12.90457119397648, 13.52553773024894, 13.77135387009324, 14.60828201917795, 14.79096888006822, 15.65673704318561, 16.04785874632436
