L(s) = 1 | + 3-s + 3·7-s − 2·9-s − 3·13-s + 3·17-s − 19-s + 3·21-s − 9·23-s − 5·25-s − 5·27-s + 9·29-s − 6·31-s − 6·37-s − 3·39-s − 6·41-s − 8·43-s + 2·49-s + 3·51-s + 9·53-s − 57-s − 3·59-s − 6·61-s − 6·63-s + 5·67-s − 9·69-s − 11·73-s − 5·75-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s − 0.832·13-s + 0.727·17-s − 0.229·19-s + 0.654·21-s − 1.87·23-s − 25-s − 0.962·27-s + 1.67·29-s − 1.07·31-s − 0.986·37-s − 0.480·39-s − 0.937·41-s − 1.21·43-s + 2/7·49-s + 0.420·51-s + 1.23·53-s − 0.132·57-s − 0.390·59-s − 0.768·61-s − 0.755·63-s + 0.610·67-s − 1.08·69-s − 1.28·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−8.077318389877249878575568705166, −7.45188899568233095526898184544, −6.47245777602371708280434034405, −5.57923734076542211171940816363, −5.02441970180507927747644335160, −4.10160942656827275705442362521, −3.30797649842366399682376003332, −2.28869671875634074476725481709, −1.66328132033866346397649251640, 0,
1.66328132033866346397649251640, 2.28869671875634074476725481709, 3.30797649842366399682376003332, 4.10160942656827275705442362521, 5.02441970180507927747644335160, 5.57923734076542211171940816363, 6.47245777602371708280434034405, 7.45188899568233095526898184544, 8.077318389877249878575568705166