L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.965 + 0.258i)5-s − 0.999i·8-s + (0.5 − 0.866i)9-s + (0.965 + 0.258i)10-s + 1.41i·13-s + (−0.5 + 0.866i)16-s + (1.22 − 0.707i)17-s + (−0.866 + 0.499i)18-s + (−0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + (0.707 − 1.22i)26-s + (0.866 − 0.499i)32-s − 1.41·34-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.965 + 0.258i)5-s − 0.999i·8-s + (0.5 − 0.866i)9-s + (0.965 + 0.258i)10-s + 1.41i·13-s + (−0.5 + 0.866i)16-s + (1.22 − 0.707i)17-s + (−0.866 + 0.499i)18-s + (−0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + (0.707 − 1.22i)26-s + (0.866 − 0.499i)32-s − 1.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6215799039\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6215799039\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.965 - 0.258i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.41iT - T^{2} \) |
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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
−9.922552197709276684957316387725, −9.493600949540969785583798795577, −8.583758314404481453244171368113, −7.67021436107121208520001598496, −7.06652642881219069541522391606, −6.25535783319026214325310758899, −4.48927676494909329218206359569, −3.74268858308021409640504315515, −2.73960846638942985615717002133, −1.12503455848332174329274823098,
1.09103090404951139878806406777, 2.72931873858675939607097594529, 4.11615482734074465580480162201, 5.24837717668583242438779797325, 5.97175471216079513764284289979, 7.35254388748331778906011670772, 7.77847901556412539578274787496, 8.299000091668864983571778507251, 9.350958932551236198169326493779, 10.31039097304477237609774471422