L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.41 − i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−1.70 − 0.292i)6-s + 4·7-s + (0.707 − 0.707i)8-s + (1.00 − 2.82i)9-s + 1.00·10-s − 4.24·11-s + (1.00 + 1.41i)12-s + (−1 − i)13-s + (−2.82 − 2.82i)14-s + (−0.292 + 1.70i)15-s − 1.00·16-s + (1.41 − 1.41i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.816 − 0.577i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.696 − 0.119i)6-s + 1.51·7-s + (0.250 − 0.250i)8-s + (0.333 − 0.942i)9-s + 0.316·10-s − 1.27·11-s + (0.288 + 0.408i)12-s + (−0.277 − 0.277i)13-s + (−0.755 − 0.755i)14-s + (−0.0756 + 0.440i)15-s − 0.250·16-s + (0.342 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0872 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0872 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.689750394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.689750394\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.41 + i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (-6 + i)T \) |
good | 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + (2 + 2i)T + 19iT^{2} \) |
| 23 | \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \) |
| 29 | \( 1 + 29iT^{2} \) |
| 31 | \( 1 + (-5 + 5i)T - 31iT^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + (-9 - 9i)T + 43iT^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 2.82iT - 53T^{2} \) |
| 59 | \( 1 + (-2.82 + 2.82i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4 + 4i)T - 61iT^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + (3 + 3i)T + 79iT^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + (-12.7 - 12.7i)T + 89iT^{2} \) |
| 97 | \( 1 + (12 + 12i)T + 97iT^{2} \) |
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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
−9.579064423965869101406996895641, −8.603085466394418220448473421810, −7.913654311374382653214691632121, −7.67475714634556260722835862912, −6.58832818291790810094777898682, −5.12421791754311686281628464165, −4.24498609967899558218797644174, −2.82989870082660568266227677528, −2.30196824802098305971659635939, −0.861569195168630739258680216513,
1.50009799207778545486043385307, 2.67021157358878932183785255397, 4.11857235334476121743580824559, 4.95229467080386844360094903867, 5.53208497836927928756840890794, 7.19161426308771528526955445868, 7.84081320725058623279873879045, 8.342828308349547007918214724953, 8.975582050782904072151136238992, 10.01557331369368587447359057125