| L(s) = 1 | − 8.12·2-s + 34.0·4-s − 17.4·5-s − 16.5·8-s + 141.·10-s + 114.·11-s − 205.·13-s − 954.·16-s + 757.·17-s − 1.01e3·19-s − 592.·20-s − 928.·22-s + 916.·23-s − 2.82e3·25-s + 1.66e3·26-s + 1.09e3·29-s + 8.23e3·31-s + 8.28e3·32-s − 6.15e3·34-s − 1.07e4·37-s + 8.23e3·38-s + 287.·40-s − 1.87e4·41-s − 4.64e3·43-s + 3.88e3·44-s − 7.44e3·46-s + 1.39e4·47-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.06·4-s − 0.311·5-s − 0.0912·8-s + 0.447·10-s + 0.284·11-s − 0.336·13-s − 0.932·16-s + 0.635·17-s − 0.644·19-s − 0.331·20-s − 0.409·22-s + 0.361·23-s − 0.902·25-s + 0.483·26-s + 0.241·29-s + 1.53·31-s + 1.43·32-s − 0.912·34-s − 1.28·37-s + 0.925·38-s + 0.0284·40-s − 1.74·41-s − 0.382·43-s + 0.302·44-s − 0.518·46-s + 0.922·47-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 | \( 1 + 8.12T + 32T^{2} \) |
| 5 | \( 1 + 17.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 114.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 205.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 757.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.01e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 916.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.87e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.64e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.99e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.06e4T + 8.58e9T^{2} \) |
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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
−9.986020396809120544915260008141, −8.821741291716433464569823895717, −8.275075147501063938340219028313, −7.32830282067222861140296078051, −6.52632119035924993493150907776, −5.09323251893312900939946692630, −3.82601017729652787981938778258, −2.32879932055782628513131329922, −1.11663841645720387559669160235, 0,
1.11663841645720387559669160235, 2.32879932055782628513131329922, 3.82601017729652787981938778258, 5.09323251893312900939946692630, 6.52632119035924993493150907776, 7.32830282067222861140296078051, 8.275075147501063938340219028313, 8.821741291716433464569823895717, 9.986020396809120544915260008141
