| L(s) = 1 | − 2.49·3-s + 99.5·5-s + 49·7-s − 236.·9-s − 292.·11-s + 163.·13-s − 248.·15-s − 1.50e3·17-s + 2.11e3·19-s − 122.·21-s + 4.58e3·23-s + 6.79e3·25-s + 1.19e3·27-s − 800.·29-s + 1.27e3·31-s + 729.·33-s + 4.87e3·35-s − 9.37e3·37-s − 406.·39-s + 6.58e3·41-s + 4.93e3·43-s − 2.35e4·45-s + 2.58e4·47-s + 2.40e3·49-s + 3.76e3·51-s + 6.21e3·53-s − 2.91e4·55-s + ⋯ |
| L(s) = 1 | − 0.159·3-s + 1.78·5-s + 0.377·7-s − 0.974·9-s − 0.729·11-s + 0.267·13-s − 0.284·15-s − 1.26·17-s + 1.34·19-s − 0.0604·21-s + 1.80·23-s + 2.17·25-s + 0.315·27-s − 0.176·29-s + 0.238·31-s + 0.116·33-s + 0.673·35-s − 1.12·37-s − 0.0428·39-s + 0.611·41-s + 0.407·43-s − 1.73·45-s + 1.70·47-s + 0.142·49-s + 0.202·51-s + 0.303·53-s − 1.29·55-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
| \(L(3)\) |
\(\approx\) |
\(2.847794188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.847794188\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.49T + 243T^{2} \) |
| 5 | \( 1 - 99.5T + 3.12e3T^{2} \) |
| 11 | \( 1 + 292.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 163.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.50e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 800.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.27e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.37e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.58e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.21e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.80e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 828.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.28e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e5T + 8.58e9T^{2} \) |
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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
−10.47669045362204087429389385256, −9.149656359513156739898005296323, −8.911750465026696434840087713693, −7.44174453711612426404354178027, −6.36466760917782045943216727952, −5.50459383331174138363660490966, −4.93646220555092196888802000276, −3.00131583837429881954558577678, −2.17092900144961024919154122304, −0.895179071032358757109278635352,
0.895179071032358757109278635352, 2.17092900144961024919154122304, 3.00131583837429881954558577678, 4.93646220555092196888802000276, 5.50459383331174138363660490966, 6.36466760917782045943216727952, 7.44174453711612426404354178027, 8.911750465026696434840087713693, 9.149656359513156739898005296323, 10.47669045362204087429389385256
