| L(s) = 1 | − 1.41i·2-s + 3.90·3-s − 2.00·4-s + 2.23·5-s − 5.52i·6-s + 6.10i·7-s + 2.82i·8-s + 6.27·9-s − 3.16i·10-s − 7.81·12-s + 17.0i·13-s + 8.63·14-s + 8.73·15-s + 4.00·16-s + 8.77i·17-s − 8.87i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.30·3-s − 0.500·4-s + 0.447·5-s − 0.921i·6-s + 0.872i·7-s + 0.353i·8-s + 0.697·9-s − 0.316i·10-s − 0.651·12-s + 1.31i·13-s + 0.616·14-s + 0.582·15-s + 0.250·16-s + 0.516i·17-s − 0.493i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.967217949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.967217949\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 - 2.23T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 3.90T + 9T^{2} \) |
| 7 | \( 1 - 6.10iT - 49T^{2} \) |
| 13 | \( 1 - 17.0iT - 169T^{2} \) |
| 17 | \( 1 - 8.77iT - 289T^{2} \) |
| 19 | \( 1 + 19.0iT - 361T^{2} \) |
| 23 | \( 1 - 7.59T + 529T^{2} \) |
| 29 | \( 1 - 47.7iT - 841T^{2} \) |
| 31 | \( 1 + 0.442T + 961T^{2} \) |
| 37 | \( 1 - 49.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 45.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.14iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 66.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 96.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 8.84T + 3.48e3T^{2} \) |
| 61 | \( 1 - 54.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 66.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 48.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 0.643iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 87.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 134. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 70.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 0.284T + 9.40e3T^{2} \) |
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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
−9.314375875184677701514330668862, −8.978306733476086668873134318879, −8.401966817712505371770072918613, −7.27406205190755390713458963503, −6.27996197507285147428707750065, −5.13226324921491298321885612934, −4.16702254267069364444336669709, −3.06344773109161610015593891184, −2.40024335503414436505745269940, −1.50905315154676807468175909186,
0.74511382485739363037939836238, 2.30494888116930105656517844011, 3.35830759226117364551433064789, 4.16040013601561150426741468354, 5.35342508897519062154077886653, 6.20949382732159966588634149384, 7.33706225495875531130322772576, 7.87059576416179408568274975647, 8.460420610052096958185348170857, 9.455385413626359272690693828277
