| L(s) = 1 | − 2.72·2-s + (−1.44 − 0.948i)3-s + 5.40·4-s − i·5-s + (3.94 + 2.58i)6-s − i·7-s − 9.27·8-s + (1.20 + 2.74i)9-s + 2.72i·10-s + (−3.31 − 0.00239i)11-s + (−7.83 − 5.12i)12-s − 0.172i·13-s + 2.72i·14-s + (−0.948 + 1.44i)15-s + 14.4·16-s + 0.805·17-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + (−0.836 − 0.547i)3-s + 2.70·4-s − 0.447i·5-s + (1.61 + 1.05i)6-s − 0.377i·7-s − 3.27·8-s + (0.400 + 0.916i)9-s + 0.860i·10-s + (−0.999 − 0.000720i)11-s + (−2.26 − 1.47i)12-s − 0.0477i·13-s + 0.727i·14-s + (−0.244 + 0.374i)15-s + 3.60·16-s + 0.195·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3961218771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3961218771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.44 + 0.948i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (3.31 + 0.00239i)T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 13 | \( 1 + 0.172iT - 13T^{2} \) |
| 17 | \( 1 - 0.805T + 17T^{2} \) |
| 19 | \( 1 - 2.99iT - 19T^{2} \) |
| 23 | \( 1 - 0.277iT - 23T^{2} \) |
| 29 | \( 1 - 7.75T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 - 9.92iT - 43T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 5.52iT - 59T^{2} \) |
| 61 | \( 1 + 4.15iT - 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 - 14.8iT - 71T^{2} \) |
| 73 | \( 1 + 6.17iT - 73T^{2} \) |
| 79 | \( 1 + 3.40iT - 79T^{2} \) |
| 83 | \( 1 - 18.1T + 83T^{2} \) |
| 89 | \( 1 + 14.7iT - 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−9.991955296162742634020502473719, −8.791813594102940901671549724457, −7.942784222344447162102711714449, −7.64111660787110217026278311624, −6.59461209599774726393572483964, −5.94863758600427054700739555857, −4.80780883037923925810842950218, −2.89333579098096168878347924000, −1.67887578787855147271759645734, −0.67820460825044273674017414708,
0.62452898512757236969029910500, 2.23431355660036754129242853185, 3.25828574258978566266676152185, 4.99438441145016205073275234585, 5.99954496316578934971018277033, 6.74472878203124199415693697985, 7.48156893151877024681452209249, 8.451397765739935025121199924853, 9.113345183442214910191697112748, 10.03751967968687792701067653046
