| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 4·11-s − 5·13-s − 15-s − 4·17-s + 19-s − 21-s − 23-s − 4·25-s + 5·27-s + 4·31-s + 4·33-s + 35-s + 10·37-s + 5·39-s + 6·41-s − 8·43-s − 2·45-s − 4·47-s + 49-s + 4·51-s + 8·53-s − 4·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.20·11-s − 1.38·13-s − 0.258·15-s − 0.970·17-s + 0.229·19-s − 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.962·27-s + 0.718·31-s + 0.696·33-s + 0.169·35-s + 1.64·37-s + 0.800·39-s + 0.937·41-s − 1.21·43-s − 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.09·53-s − 0.539·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.89640559702898, −14.32721371708640, −13.80313336289198, −13.25491867109501, −12.87282338069549, −12.19719554335353, −11.63990264510841, −11.36186488269093, −10.70704310410659, −10.16738353169468, −9.772938010352427, −9.170680433910475, −8.454609687462373, −7.932042178317187, −7.521444569746741, −6.736858381278631, −6.225886790672433, −5.574355406373187, −5.154300417921431, −4.682553121270946, −4.011408777197177, −2.886009021965268, −2.504277193849026, −1.955711746251645, −0.7435015331029543, 0,
0.7435015331029543, 1.955711746251645, 2.504277193849026, 2.886009021965268, 4.011408777197177, 4.682553121270946, 5.154300417921431, 5.574355406373187, 6.225886790672433, 6.736858381278631, 7.521444569746741, 7.932042178317187, 8.454609687462373, 9.170680433910475, 9.772938010352427, 10.16738353169468, 10.70704310410659, 11.36186488269093, 11.63990264510841, 12.19719554335353, 12.87282338069549, 13.25491867109501, 13.80313336289198, 14.32721371708640, 14.89640559702898
