| L(s) = 1 | + 3·3-s − 15.3·5-s − 7·7-s + 9·9-s + 5.35·11-s − 11.2·13-s − 46.0·15-s + 94.0·17-s − 20·19-s − 21·21-s + 102.·23-s + 110.·25-s + 27·27-s − 102·29-s + 341.·31-s + 16.0·33-s + 107.·35-s − 288.·37-s − 33.8·39-s − 252.·41-s + 145.·43-s − 138.·45-s − 573.·47-s + 49·49-s + 282.·51-s + 234.·53-s − 82.2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.37·5-s − 0.377·7-s + 0.333·9-s + 0.146·11-s − 0.240·13-s − 0.793·15-s + 1.34·17-s − 0.241·19-s − 0.218·21-s + 0.931·23-s + 0.886·25-s + 0.192·27-s − 0.653·29-s + 1.97·31-s + 0.0847·33-s + 0.519·35-s − 1.28·37-s − 0.139·39-s − 0.963·41-s + 0.516·43-s − 0.457·45-s − 1.78·47-s + 0.142·49-s + 0.774·51-s + 0.607·53-s − 0.201·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 + 15.3T + 125T^{2} \) |
| 11 | \( 1 - 5.35T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 94.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 102T + 2.43e4T^{2} \) |
| 31 | \( 1 - 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 573.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 65.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 380.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 469.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.55e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.611782201899371217610334746497, −8.101400152727565185947998120773, −7.31555490090488496098742203100, −6.64217499580888302623195553221, −5.34064764440222788612520558447, −4.36607747138974483316800695582, −3.50660441317413794846140365007, −2.86857453650217487667455940126, −1.27404936513904318896815550117, 0,
1.27404936513904318896815550117, 2.86857453650217487667455940126, 3.50660441317413794846140365007, 4.36607747138974483316800695582, 5.34064764440222788612520558447, 6.64217499580888302623195553221, 7.31555490090488496098742203100, 8.101400152727565185947998120773, 8.611782201899371217610334746497
