| L(s) = 1 | − 4i·2-s − 16·4-s − 170i·7-s + 64i·8-s + 243·9-s − 250·11-s + 169i·13-s − 680·14-s + 256·16-s + 1.06e3i·17-s − 972i·18-s + 78·19-s + 1.00e3i·22-s − 1.57e3i·23-s + 676·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.31i·7-s + 0.353i·8-s + 9-s − 0.622·11-s + 0.277i·13-s − 0.927·14-s + 0.250·16-s + 0.891i·17-s − 0.707i·18-s + 0.0495·19-s + 0.440i·22-s − 0.621i·23-s + 0.196·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6100755993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6100755993\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 3 | \( 1 - 243T^{2} \) |
| 7 | \( 1 + 170iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 250T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.06e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 78T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.57e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.09e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.05e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.90e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.96e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.90e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.95e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.67e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.74e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.82e5iT - 8.58e9T^{2} \) |
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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
−10.20342167150840282451778053937, −9.327414086266412821706191588042, −8.156622463232810170321130873854, −7.38754431193185287615937658385, −6.49492079448778504743548669428, −5.09010932719553827065645751240, −4.18269143092090988590227267580, −3.48944317235194583476508427491, −1.99381650084970716669695554880, −1.06935338525208003745537233715,
0.13907298106046146538449847430, 1.73768026712978313587296239269, 2.94276659651090590896011864796, 4.24330809710752194713377085436, 5.36825008280670522608614138551, 5.81876445239945086251291291849, 7.19458845854385260263934825074, 7.62842051931812832181152162594, 8.881311486413362419027762561393, 9.338332789163391110963464133790
