| L(s) = 1 | − 5.44·2-s − 2.33·4-s + 36·5-s + 187.·8-s − 196.·10-s − 184.·11-s − 147.·13-s − 943.·16-s + 1.96e3·17-s − 1.89e3·19-s − 84.1·20-s + 1.00e3·22-s − 136.·23-s − 1.82e3·25-s + 805.·26-s + 1.25e3·29-s − 8.96e3·31-s − 844.·32-s − 1.07e4·34-s + 1.28e4·37-s + 1.03e4·38-s + 6.73e3·40-s + 8.97e3·41-s + 1.35e4·43-s + 431.·44-s + 746.·46-s − 2.00e4·47-s + ⋯ |
| L(s) = 1 | − 0.962·2-s − 0.0730·4-s + 0.643·5-s + 1.03·8-s − 0.620·10-s − 0.459·11-s − 0.242·13-s − 0.921·16-s + 1.65·17-s − 1.20·19-s − 0.0470·20-s + 0.442·22-s − 0.0539·23-s − 0.585·25-s + 0.233·26-s + 0.278·29-s − 1.67·31-s − 0.145·32-s − 1.59·34-s + 1.54·37-s + 1.15·38-s + 0.665·40-s + 0.833·41-s + 1.11·43-s + 0.0335·44-s + 0.0519·46-s − 1.32·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.44T + 32T^{2} \) |
| 5 | \( 1 - 36T + 3.12e3T^{2} \) |
| 11 | \( 1 + 184.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 147.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.96e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.89e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 136.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.96e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.33e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.86e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.18e5T + 8.58e9T^{2} \) |
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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
−9.804401555674077473151487803108, −9.117211541397892695750235360660, −8.059651129770088427444307311994, −7.46683412089697881603832390900, −6.10807044401003171986234528115, −5.17254374495649164973177920562, −3.94901226716377999395078046754, −2.38465726980938027495563504542, −1.25082600036361176064315954190, 0,
1.25082600036361176064315954190, 2.38465726980938027495563504542, 3.94901226716377999395078046754, 5.17254374495649164973177920562, 6.10807044401003171986234528115, 7.46683412089697881603832390900, 8.059651129770088427444307311994, 9.117211541397892695750235360660, 9.804401555674077473151487803108
