| L(s) = 1 | + (0.754 + 2.32i)2-s + (0.809 − 0.587i)3-s + (−3.19 + 2.32i)4-s + (0.824 − 2.07i)5-s + (1.97 + 1.43i)6-s − 3.44·7-s + (−3.85 − 2.80i)8-s + (0.309 − 0.951i)9-s + (5.44 + 0.345i)10-s + (−1.00 − 3.10i)11-s + (−1.22 + 3.76i)12-s + (−0.998 + 3.07i)13-s + (−2.59 − 7.98i)14-s + (−0.554 − 2.16i)15-s + (1.15 − 3.54i)16-s + (4.08 + 2.97i)17-s + ⋯ |
| L(s) = 1 | + (0.533 + 1.64i)2-s + (0.467 − 0.339i)3-s + (−1.59 + 1.16i)4-s + (0.368 − 0.929i)5-s + (0.805 + 0.585i)6-s − 1.30·7-s + (−1.36 − 0.991i)8-s + (0.103 − 0.317i)9-s + (1.72 + 0.109i)10-s + (−0.304 − 0.936i)11-s + (−0.352 + 1.08i)12-s + (−0.277 + 0.852i)13-s + (−0.693 − 2.13i)14-s + (−0.143 − 0.559i)15-s + (0.288 − 0.887i)16-s + (0.991 + 0.720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0638 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0638 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.871391 + 0.817413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.871391 + 0.817413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.824 + 2.07i)T \) |
| good | 2 | \( 1 + (-0.754 - 2.32i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + (1.00 + 3.10i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.998 - 3.07i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.08 - 2.97i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.49 - 1.81i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.478 - 1.47i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.52 + 1.83i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (6.02 + 4.37i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.77 - 5.47i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.67 + 5.15i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.53T + 43T^{2} \) |
| 47 | \( 1 + (5.72 - 4.15i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.21 + 5.96i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.534 + 1.64i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.42 - 7.45i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.49 - 1.08i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.577 + 0.419i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.581 + 1.78i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-10.7 + 7.83i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.20 + 2.32i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.63 - 8.11i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.61 + 6.26i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.77236012280791390501250414635, −13.73965546883758600370690429221, −13.17759179608132725519060310933, −12.22609151666676169426979045347, −9.743716589011784273303222004972, −8.774190534160794397404216357303, −7.72080196605930933339695191727, −6.39411961991546874057009950072, −5.49783599524686430250853329017, −3.72896151367082611112736316458,
2.65124970178725384295663068678, 3.45658104320446481934489014460, 5.28161689370155824428026610955, 7.19643679874577352931297938842, 9.462897234190995462728609543445, 9.986227741395356783120552370673, 10.80309005592240004893268976244, 12.25808599293263495774418662169, 13.02439847913972101104251860120, 14.01503832119550743541200312809
