| L(s) = 1 | − 3·3-s − 5·5-s − 19.9·7-s + 9·9-s − 27.0·11-s + 47.3·13-s + 15·15-s − 73.0·17-s + 158.·19-s + 59.8·21-s − 23·23-s + 25·25-s − 27·27-s + 70.1·29-s − 12.6·31-s + 81.1·33-s + 99.8·35-s + 338.·37-s − 141.·39-s − 361.·41-s + 328.·43-s − 45·45-s + 251.·47-s + 55.4·49-s + 219.·51-s + 180.·53-s + 135.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.07·7-s + 0.333·9-s − 0.741·11-s + 1.00·13-s + 0.258·15-s − 1.04·17-s + 1.90·19-s + 0.622·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.449·29-s − 0.0732·31-s + 0.428·33-s + 0.482·35-s + 1.50·37-s − 0.582·39-s − 1.37·41-s + 1.16·43-s − 0.149·45-s + 0.780·47-s + 0.161·49-s + 0.601·51-s + 0.468·53-s + 0.331·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 7 | \( 1 + 19.9T + 343T^{2} \) |
| 11 | \( 1 + 27.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 47.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 158.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 70.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 328.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 206.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 802.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 112.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 174.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 716.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 94.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
| 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
−8.908560094265564639780458606309, −7.88461161030701786613253571169, −7.12750712730242106693671623326, −6.28400453506498676079288120723, −5.59380216319041188080643941651, −4.55599227057851867907257537204, −3.57402411004796810510130068070, −2.70162570055469628002491575244, −1.06884933768544897699140276657, 0,
1.06884933768544897699140276657, 2.70162570055469628002491575244, 3.57402411004796810510130068070, 4.55599227057851867907257537204, 5.59380216319041188080643941651, 6.28400453506498676079288120723, 7.12750712730242106693671623326, 7.88461161030701786613253571169, 8.908560094265564639780458606309
