| L(s) = 1 | − 10.2·2-s + 72.5·4-s + 23.7·5-s − 414.·8-s − 242.·10-s − 465.·11-s + 1.01e3·13-s + 1.91e3·16-s + 561.·17-s + 1.38e3·19-s + 1.72e3·20-s + 4.75e3·22-s − 4.11e3·23-s − 2.56e3·25-s − 1.04e4·26-s + 2.38e3·29-s − 2.95e3·31-s − 6.32e3·32-s − 5.74e3·34-s − 9.90e3·37-s − 1.41e4·38-s − 9.84e3·40-s − 4.47e3·41-s + 5.18e3·43-s − 3.37e4·44-s + 4.20e4·46-s − 3.12e3·47-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 2.26·4-s + 0.424·5-s − 2.28·8-s − 0.767·10-s − 1.16·11-s + 1.67·13-s + 1.87·16-s + 0.471·17-s + 0.881·19-s + 0.963·20-s + 2.09·22-s − 1.62·23-s − 0.819·25-s − 3.02·26-s + 0.525·29-s − 0.551·31-s − 1.09·32-s − 0.852·34-s − 1.18·37-s − 1.59·38-s − 0.972·40-s − 0.415·41-s + 0.427·43-s − 2.62·44-s + 2.93·46-s − 0.206·47-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 | \( 1 + 10.2T + 32T^{2} \) |
| 5 | \( 1 - 23.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 465.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.01e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 561.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.38e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.14e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.07e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.84e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.36e4T + 8.58e9T^{2} \) |
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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
−9.967312020228388171208725486010, −8.913264919404098623274643597889, −8.168418243588826338801362631954, −7.50895927152370996995303921914, −6.31797236858046959301266317514, −5.52798761132708362481698385447, −3.51955514777056648256563898116, −2.20264704818747660435119816144, −1.23082071052071592121052213364, 0,
1.23082071052071592121052213364, 2.20264704818747660435119816144, 3.51955514777056648256563898116, 5.52798761132708362481698385447, 6.31797236858046959301266317514, 7.50895927152370996995303921914, 8.168418243588826338801362631954, 8.913264919404098623274643597889, 9.967312020228388171208725486010
