L(s) = 1 | + (−1.93 − 0.5i)2-s + (3.50 + 1.93i)4-s − 5i·5-s + (−5.80 − 5.50i)8-s + (−2.5 + 9.68i)10-s + (8.50 + 13.5i)16-s − 14i·17-s − 30.9i·19-s + (9.68 − 17.5i)20-s − 30.9·23-s − 25·25-s − 61.9i·31-s + (−9.68 − 30.5i)32-s + (−7 + 27.1i)34-s + (−15.4 + 60.0i)38-s + ⋯ |
L(s) = 1 | + (−0.968 − 0.250i)2-s + (0.875 + 0.484i)4-s − i·5-s + (−0.726 − 0.687i)8-s + (−0.250 + 0.968i)10-s + (0.531 + 0.847i)16-s − 0.823i·17-s − 1.63i·19-s + (0.484 − 0.875i)20-s − 1.34·23-s − 25-s − 1.99i·31-s + (−0.302 − 0.953i)32-s + (−0.205 + 0.797i)34-s + (−0.407 + 1.57i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.484 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.380434 - 0.645269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.380434 - 0.645269i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.93 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5iT \) |
good | 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 14iT - 289T^{2} \) |
| 19 | \( 1 + 30.9iT - 361T^{2} \) |
| 23 | \( 1 + 30.9T + 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 + 61.9iT - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 92.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 86iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 118T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 61.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−11.89341782707607827539098145182, −11.17017430144331246600640766473, −9.823283701104276100675020113378, −9.170748296611558363720919005926, −8.191772253089824194086351723127, −7.19612007305331832428950264862, −5.78794047065834910712782675606, −4.24590222009882136597292985208, −2.38899370000382688648787484083, −0.59616570877129900726359977756,
1.91892529519781189932154374780, 3.55505473700686498126572765941, 5.73481952696108318749166615691, 6.62451444504420816018397345711, 7.71511614337214403637906438029, 8.577556139829743073259236462164, 10.04689154720564641590874989529, 10.40192219566774542786132363234, 11.54034235928798549631021202882, 12.46254535910019317292632806460