| L(s) = 1 | − 3·3-s + 5·5-s + 7·7-s + 9·9-s + 19.0·11-s − 2.93·13-s − 15·15-s − 6.49·17-s + 5.43·19-s − 21·21-s − 49.3·23-s + 25·25-s − 27·27-s − 291.·29-s − 244.·31-s − 57.1·33-s + 35·35-s − 193.·37-s + 8.81·39-s + 315.·41-s + 300.·43-s + 45·45-s − 86.5·47-s + 49·49-s + 19.4·51-s + 509.·53-s + 95.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.522·11-s − 0.0626·13-s − 0.258·15-s − 0.0927·17-s + 0.0656·19-s − 0.218·21-s − 0.447·23-s + 0.200·25-s − 0.192·27-s − 1.86·29-s − 1.41·31-s − 0.301·33-s + 0.169·35-s − 0.858·37-s + 0.0361·39-s + 1.20·41-s + 1.06·43-s + 0.149·45-s − 0.268·47-s + 0.142·49-s + 0.0535·51-s + 1.32·53-s + 0.233·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 11 | \( 1 - 19.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.93T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.49T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.43T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 244.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 193.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 300.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 86.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 509.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 83.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 5.25T + 2.26e5T^{2} \) |
| 67 | \( 1 + 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 863.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 326.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3T + 9.12e5T^{2} \) |
| 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
−8.800226128736525547367048934905, −7.57378765880541009769059187599, −7.07942319180339005815460755850, −5.91751534112714576786078451257, −5.55463216428386302509866682463, −4.45012538225938357165283539933, −3.63574084068821743018295982457, −2.22294046176547725604082852294, −1.34009224744706401326546983445, 0,
1.34009224744706401326546983445, 2.22294046176547725604082852294, 3.63574084068821743018295982457, 4.45012538225938357165283539933, 5.55463216428386302509866682463, 5.91751534112714576786078451257, 7.07942319180339005815460755850, 7.57378765880541009769059187599, 8.800226128736525547367048934905
