| L(s) = 1 | − 1.88i·5-s − 10.9·7-s + 0.171i·11-s − 6.48·13-s − 27.2i·17-s + 5.32·19-s + 3.51i·23-s + 21.4·25-s − 34.8i·29-s + 26.9·31-s + 20.6i·35-s − 46.4·37-s + 23.5i·41-s − 55.1·43-s + 57.5i·47-s + ⋯ |
| L(s) = 1 | − 0.376i·5-s − 1.56·7-s + 0.0155i·11-s − 0.498·13-s − 1.60i·17-s + 0.280·19-s + 0.152i·23-s + 0.858·25-s − 1.20i·29-s + 0.869·31-s + 0.590i·35-s − 1.25·37-s + 0.573i·41-s − 1.28·43-s + 1.22i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6194515825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6194515825\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.88iT - 25T^{2} \) |
| 7 | \( 1 + 10.9T + 49T^{2} \) |
| 11 | \( 1 - 0.171iT - 121T^{2} \) |
| 13 | \( 1 + 6.48T + 169T^{2} \) |
| 17 | \( 1 + 27.2iT - 289T^{2} \) |
| 19 | \( 1 - 5.32T + 361T^{2} \) |
| 23 | \( 1 - 3.51iT - 529T^{2} \) |
| 29 | \( 1 + 34.8iT - 841T^{2} \) |
| 31 | \( 1 - 26.9T + 961T^{2} \) |
| 37 | \( 1 + 46.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 23.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 57.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 82.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 79.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 44.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 41.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 66.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 115.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 36.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 158. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 62.8T + 9.40e3T^{2} \) |
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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
−9.498141357412915858067904980429, −8.713593375323496350520798338821, −7.63992348897706643360237645886, −6.90667226607802831465506446398, −6.22475327012716664086318389010, −5.23207678233358058016100160765, −4.42162883964275345247348571717, −3.21106423073226073935993043075, −2.61495549885667781636987101025, −0.905458968056834117406876203321,
0.20046215263536593428537172277, 1.83759226778075460502412433281, 3.14187255023324989744091649832, 3.56577097622615782659030517623, 4.84867559568683024634359551362, 5.84486848015364005807538846422, 6.69532648488091081703855903379, 7.02445635545170741798121185430, 8.293279407820810008174527606545, 8.907593071544861709091322188080
