L(s) = 1 | − 1.17i·2-s + (−0.707 − 0.707i)3-s + 0.624·4-s + (2.72 + 2.72i)5-s + (−0.829 + 0.829i)6-s + (−3.25 + 3.25i)7-s − 3.07i·8-s + 1.00i·9-s + (3.20 − 3.20i)10-s + (−2.99 + 2.99i)11-s + (−0.441 − 0.441i)12-s + 13-s + (3.81 + 3.81i)14-s − 3.85i·15-s − 2.36·16-s + (1.57 + 3.80i)17-s + ⋯ |
L(s) = 1 | − 0.829i·2-s + (−0.408 − 0.408i)3-s + 0.312·4-s + (1.22 + 1.22i)5-s + (−0.338 + 0.338i)6-s + (−1.22 + 1.22i)7-s − 1.08i·8-s + 0.333i·9-s + (1.01 − 1.01i)10-s + (−0.903 + 0.903i)11-s + (−0.127 − 0.127i)12-s + 0.277·13-s + (1.01 + 1.01i)14-s − 0.996i·15-s − 0.590·16-s + (0.383 + 0.923i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36656 + 0.360403i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36656 + 0.360403i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-1.57 - 3.80i)T \) |
good | 2 | \( 1 + 1.17iT - 2T^{2} \) |
| 5 | \( 1 + (-2.72 - 2.72i)T + 5iT^{2} \) |
| 7 | \( 1 + (3.25 - 3.25i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.99 - 2.99i)T - 11iT^{2} \) |
| 19 | \( 1 + 0.496iT - 19T^{2} \) |
| 23 | \( 1 + (3.26 - 3.26i)T - 23iT^{2} \) |
| 29 | \( 1 + (3.34 + 3.34i)T + 29iT^{2} \) |
| 31 | \( 1 + (-0.142 - 0.142i)T + 31iT^{2} \) |
| 37 | \( 1 + (-6.91 - 6.91i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.692 + 0.692i)T - 41iT^{2} \) |
| 43 | \( 1 - 3.68iT - 43T^{2} \) |
| 47 | \( 1 - 8.95T + 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 1.69iT - 59T^{2} \) |
| 61 | \( 1 + (-6.55 + 6.55i)T - 61iT^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + (8.77 + 8.77i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.90 - 3.90i)T + 73iT^{2} \) |
| 79 | \( 1 + (-4.59 + 4.59i)T - 79iT^{2} \) |
| 83 | \( 1 + 6.56iT - 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 + (-7.39 - 7.39i)T + 97iT^{2} \) |
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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
−10.50538935620893014776451962331, −9.906860838109302543948298981837, −9.474376535149423261046725336857, −7.81677589508858848862815131807, −6.70796329542598260039435509383, −6.20639997844079794213081623330, −5.52478617061287754962408370644, −3.48068237900842219064442401916, −2.50538997366576067212088969835, −1.98713266160611067860058584354,
0.75141699147746830948492580142, 2.66031106898590940586222744366, 4.15509784306338606160603786747, 5.45696150973672924545333471005, 5.78975605477097373391009178345, 6.72789895467142293382066005185, 7.69343521273939850056084863562, 8.775093531240671304785933978598, 9.580620784715043694645022053223, 10.36337220662536763321925215173