| L(s) = 1 | − 1.41i·2-s − 3.16·3-s − 2.00·4-s + 4.47i·6-s + (−1.58 + 2.12i)7-s + 2.82i·8-s + 7.00·9-s + 6.32·12-s + (3 + 2.23i)14-s + 4.00·16-s − 9.89i·18-s + (5.00 − 6.70i)21-s − 1.41i·23-s − 8.94i·24-s − 12.6·27-s + (3.16 − 4.24i)28-s + ⋯ |
| L(s) = 1 | − 0.999i·2-s − 1.82·3-s − 1.00·4-s + 1.82i·6-s + (−0.597 + 0.801i)7-s + 1.00i·8-s + 2.33·9-s + 1.82·12-s + (0.801 + 0.597i)14-s + 1.00·16-s − 2.33i·18-s + (1.09 − 1.46i)21-s − 0.294i·23-s − 1.82i·24-s − 2.43·27-s + (0.597 − 0.801i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.194146 - 0.386851i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.194146 - 0.386851i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (1.58 - 2.12i)T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 12.7iT - 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.37000019303406421456548011307, −9.655261144547800697016527800464, −8.776613558253986238852478407619, −7.37246963304540931767122258833, −6.22257906116902132419881796496, −5.55023497119776872212327463658, −4.74723582556657862579807091987, −3.59586098472092591920773152786, −2.00451799194513212136175888359, −0.39886386868758709040746062553,
0.938202518719182674205884212551, 3.80295141419526562831522747165, 4.64418869762787848771731375762, 5.61877671562598921583293172726, 6.23986639204093891887385205012, 7.07417447812671373059665084360, 7.65331810829051910187033847671, 9.165380843372992786615315414851, 9.996216605825706520712167156752, 10.65188378498814373655541273430
