L(s) = 1 | + 1.41i·2-s + (0.554 − 2.94i)3-s − 2.00·4-s + (4.16 + 0.783i)6-s − 2.64·7-s − 2.82i·8-s + (−8.38 − 3.26i)9-s − 2.58i·11-s + (−1.10 + 5.89i)12-s + 0.180·13-s − 3.74i·14-s + 4.00·16-s + 7.25i·17-s + (4.62 − 11.8i)18-s + 21.4·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.184 − 0.982i)3-s − 0.500·4-s + (0.694 + 0.130i)6-s − 0.377·7-s − 0.353i·8-s + (−0.931 − 0.363i)9-s − 0.234i·11-s + (−0.0923 + 0.491i)12-s + 0.0138·13-s − 0.267i·14-s + 0.250·16-s + 0.426i·17-s + (0.256 − 0.658i)18-s + 1.12·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03216080382\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03216080382\) |
\(L(2)\) |
|
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 | \( 1 + (-0.554 + 2.94i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 11 | \( 1 + 2.58iT - 121T^{2} \) |
| 13 | \( 1 - 0.180T + 169T^{2} \) |
| 17 | \( 1 - 7.25iT - 289T^{2} \) |
| 19 | \( 1 - 21.4T + 361T^{2} \) |
| 23 | \( 1 + 11.8iT - 529T^{2} \) |
| 29 | \( 1 + 29.5iT - 841T^{2} \) |
| 31 | \( 1 + 49.5T + 961T^{2} \) |
| 37 | \( 1 + 43.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 20.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 74.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 7.05iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 55.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 100. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.44T + 3.72e3T^{2} \) |
| 67 | \( 1 + 85.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 97.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 17.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 3.08iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 80.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 105.T + 9.40e3T^{2} \) |
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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
−9.879772974776626206624000183499, −9.026424533113736664500426231363, −8.318017362601937186720193354374, −7.50105954227122897524318398444, −6.84903350445627998636440393519, −5.99899660761363210499455364289, −5.29043451455218407659060904579, −3.86867641400019364489840055305, −2.84239738226892349530147843202, −1.40334898769794615082367206570,
0.009684149920124463942203587922, 1.82639560899904908048630173898, 3.21483379347415430511665410851, 3.63921765150960248806510056251, 4.97535473598687636511456381024, 5.42454749149776538327975992054, 6.83804391312518090428334411317, 7.86743911103664676878172149084, 8.920141166664153362326613727692, 9.421692978163260796921402868534