| L(s) = 1 | − 1.41i·2-s + (−1.59 − 2.54i)3-s − 2.00·4-s − 2.23i·5-s + (−3.59 + 2.25i)6-s + 10.3·7-s + 2.82i·8-s + (−3.92 + 8.10i)9-s − 3.16·10-s − 14.4i·11-s + (3.18 + 5.08i)12-s + 14.5·13-s − 14.5i·14-s + (−5.68 + 3.56i)15-s + 4.00·16-s + 0.438i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.531 − 0.847i)3-s − 0.500·4-s − 0.447i·5-s + (−0.599 + 0.375i)6-s + 1.47·7-s + 0.353i·8-s + (−0.435 + 0.900i)9-s − 0.316·10-s − 1.31i·11-s + (0.265 + 0.423i)12-s + 1.12·13-s − 1.04i·14-s + (−0.378 + 0.237i)15-s + 0.250·16-s + 0.0257i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.792124478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.792124478\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (1.59 + 2.54i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + 4.79iT \) |
| good | 7 | \( 1 - 10.3T + 49T^{2} \) |
| 11 | \( 1 + 14.4iT - 121T^{2} \) |
| 13 | \( 1 - 14.5T + 169T^{2} \) |
| 17 | \( 1 - 0.438iT - 289T^{2} \) |
| 19 | \( 1 - 30.5T + 361T^{2} \) |
| 29 | \( 1 + 23.6iT - 841T^{2} \) |
| 31 | \( 1 - 37.8T + 961T^{2} \) |
| 37 | \( 1 - 3.95T + 1.36e3T^{2} \) |
| 41 | \( 1 + 0.897iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 47.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 39.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 67.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 43.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 69.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 15.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 32.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 136.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 117.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 72.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 48.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 128.T + 9.40e3T^{2} \) |
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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
−10.20981925719390202031335381558, −8.763171199405511290173670029691, −8.310862652723513540310781804363, −7.52847832409368008103111618806, −6.06732450401704904441998495345, −5.39997453081499335314333070601, −4.41938810315252604306979724778, −2.99719281704540928740028301815, −1.52115990028203470654015271232, −0.846146480351116671107964335104,
1.39029547423532614776486635058, 3.34514085138136966657764883527, 4.54913812728969085740720495907, 5.06951367354411170527030383186, 6.07446308911607043619693946076, 7.10667768309471226534686876400, 7.949627685727703295206141872659, 8.873890229163253143158504999958, 9.808622610639547791350419520510, 10.51770018917795748072523007883
