| L(s) = 1 | − 1.73i·3-s − 8.99i·5-s − 2.99·9-s − 4.63·11-s + 15.7i·13-s − 15.5·15-s − 9.25i·17-s − 27.5i·19-s + 5.65·23-s − 55.9·25-s + 5.19i·27-s − 14.6·29-s + 9.97i·31-s + 8.03i·33-s − 50.8·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.79i·5-s − 0.333·9-s − 0.421·11-s + 1.21i·13-s − 1.03·15-s − 0.544i·17-s − 1.44i·19-s + 0.245·23-s − 2.23·25-s + 0.192i·27-s − 0.506·29-s + 0.321i·31-s + 0.243i·33-s − 1.37·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6249119358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6249119358\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 8.99iT - 25T^{2} \) |
| 11 | \( 1 + 4.63T + 121T^{2} \) |
| 13 | \( 1 - 15.7iT - 169T^{2} \) |
| 17 | \( 1 + 9.25iT - 289T^{2} \) |
| 19 | \( 1 + 27.5iT - 361T^{2} \) |
| 23 | \( 1 - 5.65T + 529T^{2} \) |
| 29 | \( 1 + 14.6T + 841T^{2} \) |
| 31 | \( 1 - 9.97iT - 961T^{2} \) |
| 37 | \( 1 + 50.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 70.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 49.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 8.56T + 2.80e3T^{2} \) |
| 59 | \( 1 - 98.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 10.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 93.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 124. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 136. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 129. iT - 9.40e3T^{2} \) |
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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
−8.969509663900205708441632832530, −8.455111429468526981070203861985, −7.39261885266815239252788176664, −6.68808644258920542242172151050, −5.38257476603480866853528238260, −4.94073636728643830304724451303, −3.92540922941142062902355053394, −2.35359419665206812849644047306, −1.30724645734967346904079831716, −0.18490901017054961695467360482,
2.03846839634815980131697586523, 3.26323348054475278356994554407, 3.57491277002239266592286432030, 5.08934961356075985786078365654, 5.98715985964835843989543289570, 6.67861962170215100186192889602, 7.78125502898677776887213577594, 8.202580391938492611003291550449, 9.665346132481143069600244040138, 10.19966597852157727864711768435
