L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (−0.5 + 0.866i)11-s + (−1.22 + 1.22i)13-s + (−0.866 + 0.500i)14-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)18-s + (0.866 + 1.5i)19-s + (0.707 − 0.707i)22-s + (−0.258 + 0.965i)23-s + (1.49 − 0.866i)26-s + (0.965 − 0.258i)28-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (−0.5 + 0.866i)11-s + (−1.22 + 1.22i)13-s + (−0.866 + 0.500i)14-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)18-s + (0.866 + 1.5i)19-s + (0.707 − 0.707i)22-s + (−0.258 + 0.965i)23-s + (1.49 − 0.866i)26-s + (0.965 − 0.258i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5719567295\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5719567295\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + 1.73iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.880390038394938079558133895801, −9.258955773612922295256794386199, −8.169816396613874052430248934658, −7.58508719647693763513763242294, −7.13302604869774877239464138853, −5.84396185942796397923837095049, −4.85387813203063674462403538027, −3.79061083581996722743594145270, −2.45745800860247170163570861473, −1.63034855605937855165070787651,
0.63045122520183065317989600293, 2.53544836201763875529287668594, 2.94433072391835543899915454449, 4.98980739210842438544548205804, 5.51856783783194984091914286084, 6.37285307866907345167183001636, 7.44446388244459279016202196392, 8.162343436641977506667578916062, 8.690168023703550907243038944130, 9.479198129631908431429591203627