| L(s) = 1 | + 9.18·2-s + 52.2·4-s + 22.0·5-s + 186.·8-s + 202.·10-s − 416.·11-s − 797.·13-s + 37.3·16-s − 1.37e3·17-s − 2.31e3·19-s + 1.15e3·20-s − 3.82e3·22-s + 955.·23-s − 2.63e3·25-s − 7.32e3·26-s + 7.03e3·29-s − 1.26e3·31-s − 5.61e3·32-s − 1.26e4·34-s + 9.77e3·37-s − 2.12e4·38-s + 4.11e3·40-s − 5.40e3·41-s + 1.96e4·43-s − 2.17e4·44-s + 8.77e3·46-s + 2.05e3·47-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.63·4-s + 0.394·5-s + 1.02·8-s + 0.640·10-s − 1.03·11-s − 1.30·13-s + 0.0364·16-s − 1.15·17-s − 1.46·19-s + 0.645·20-s − 1.68·22-s + 0.376·23-s − 0.844·25-s − 2.12·26-s + 1.55·29-s − 0.235·31-s − 0.970·32-s − 1.87·34-s + 1.17·37-s − 2.38·38-s + 0.406·40-s − 0.501·41-s + 1.62·43-s − 1.69·44-s + 0.611·46-s + 0.135·47-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 | \( 1 - 9.18T + 32T^{2} \) |
| 5 | \( 1 - 22.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 416.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 797.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 955.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.40e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.96e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.05e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.43e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.49e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.76e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.44e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.93e4T + 8.58e9T^{2} \) |
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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
−10.17426835479203949860085825689, −8.975709796056224279015756613417, −7.72121912494408954143639425238, −6.68421161297924410988121168967, −5.89309023293528381147225084958, −4.85330066359067638199896035220, −4.28791827756779133440895583200, −2.73874295827328832711034695184, −2.20733745248386650151350549160, 0,
2.20733745248386650151350549160, 2.73874295827328832711034695184, 4.28791827756779133440895583200, 4.85330066359067638199896035220, 5.89309023293528381147225084958, 6.68421161297924410988121168967, 7.72121912494408954143639425238, 8.975709796056224279015756613417, 10.17426835479203949860085825689
