| L(s) = 1 | − 20.5·3-s + 53.3·5-s − 49·7-s + 178.·9-s + 90.7·11-s − 317.·13-s − 1.09e3·15-s + 797.·17-s + 264.·19-s + 1.00e3·21-s − 3.14e3·23-s − 282.·25-s + 1.32e3·27-s + 5.40e3·29-s − 6.73e3·31-s − 1.86e3·33-s − 2.61e3·35-s + 1.56e4·37-s + 6.51e3·39-s + 1.37e3·41-s − 2.26e3·43-s + 9.51e3·45-s + 1.17e4·47-s + 2.40e3·49-s − 1.63e4·51-s + 4.28e3·53-s + 4.83e3·55-s + ⋯ |
| L(s) = 1 | − 1.31·3-s + 0.953·5-s − 0.377·7-s + 0.734·9-s + 0.226·11-s − 0.521·13-s − 1.25·15-s + 0.669·17-s + 0.167·19-s + 0.497·21-s − 1.23·23-s − 0.0904·25-s + 0.350·27-s + 1.19·29-s − 1.25·31-s − 0.297·33-s − 0.360·35-s + 1.88·37-s + 0.686·39-s + 0.128·41-s − 0.186·43-s + 0.700·45-s + 0.773·47-s + 0.142·49-s − 0.881·51-s + 0.209·53-s + 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 20.5T + 243T^{2} \) |
| 5 | \( 1 - 53.3T + 3.12e3T^{2} \) |
| 11 | \( 1 - 90.7T + 1.61e5T^{2} \) |
| 13 | \( 1 + 317.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 797.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 264.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.56e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.26e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.17e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.28e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.10e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.58e5T + 8.58e9T^{2} \) |
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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
−9.993813130779695987619253244288, −9.272203210219002934324800578204, −7.86771402105331617093233754230, −6.71667049770202196481467608680, −5.93351238517063720615569676569, −5.38985463625550691061577597640, −4.17654654750206692338886244529, −2.58845152139546218401246917120, −1.22830552368028312279152911099, 0,
1.22830552368028312279152911099, 2.58845152139546218401246917120, 4.17654654750206692338886244529, 5.38985463625550691061577597640, 5.93351238517063720615569676569, 6.71667049770202196481467608680, 7.86771402105331617093233754230, 9.272203210219002934324800578204, 9.993813130779695987619253244288
