| L(s) = 1 | + 6.89·3-s + 48.4·5-s + 49·7-s − 195.·9-s − 163.·11-s + 120.·13-s + 334.·15-s + 78.1·17-s − 2.26e3·19-s + 337.·21-s + 2.45e3·23-s − 776.·25-s − 3.02e3·27-s − 6.98e3·29-s − 2.79e3·31-s − 1.12e3·33-s + 2.37e3·35-s − 9.45e3·37-s + 832.·39-s + 1.00e4·41-s − 6.93e3·43-s − 9.47e3·45-s − 1.16e3·47-s + 2.40e3·49-s + 538.·51-s − 8.56e3·53-s − 7.92e3·55-s + ⋯ |
| L(s) = 1 | + 0.442·3-s + 0.866·5-s + 0.377·7-s − 0.804·9-s − 0.407·11-s + 0.198·13-s + 0.383·15-s + 0.0656·17-s − 1.43·19-s + 0.167·21-s + 0.966·23-s − 0.248·25-s − 0.797·27-s − 1.54·29-s − 0.522·31-s − 0.180·33-s + 0.327·35-s − 1.13·37-s + 0.0876·39-s + 0.937·41-s − 0.571·43-s − 0.697·45-s − 0.0769·47-s + 0.142·49-s + 0.0290·51-s − 0.418·53-s − 0.353·55-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 | \( 1 - 6.89T + 243T^{2} \) |
| 5 | \( 1 - 48.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 163.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 120.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 78.1T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.16e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.22e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.81e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.49e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.45e5T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.742204771442840263950468195237, −8.889995195085742646118898207328, −8.219617307239260653205611173006, −7.08231500423647475505449383452, −5.92503061687771646772892221293, −5.22284757075914735261301335871, −3.82672614143940983725110317727, −2.57426549021647299323858683048, −1.71496687140387686715127508125, 0,
1.71496687140387686715127508125, 2.57426549021647299323858683048, 3.82672614143940983725110317727, 5.22284757075914735261301335871, 5.92503061687771646772892221293, 7.08231500423647475505449383452, 8.219617307239260653205611173006, 8.889995195085742646118898207328, 9.742204771442840263950468195237
