L(s) = 1 | + (5.24 − 5.24i)5-s + 5.32·7-s + (12.2 + 12.2i)11-s + (−5.73 − 5.73i)13-s + 23.3·17-s + (−11.7 + 11.7i)19-s + 5.80·23-s − 29.9i·25-s + (−18.3 − 18.3i)29-s + 16.9i·31-s + (27.9 − 27.9i)35-s + (15.3 − 15.3i)37-s + 29.2i·41-s + (−33.4 − 33.4i)43-s − 18.2i·47-s + ⋯ |
L(s) = 1 | + (1.04 − 1.04i)5-s + 0.761·7-s + (1.11 + 1.11i)11-s + (−0.441 − 0.441i)13-s + 1.37·17-s + (−0.618 + 0.618i)19-s + 0.252·23-s − 1.19i·25-s + (−0.634 − 0.634i)29-s + 0.545i·31-s + (0.798 − 0.798i)35-s + (0.414 − 0.414i)37-s + 0.713i·41-s + (−0.776 − 0.776i)43-s − 0.387i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.521108622\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.521108622\) |
\(L(2)\) |
|
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 | \( 1 + (-5.24 + 5.24i)T - 25iT^{2} \) |
| 7 | \( 1 - 5.32T + 49T^{2} \) |
| 11 | \( 1 + (-12.2 - 12.2i)T + 121iT^{2} \) |
| 13 | \( 1 + (5.73 + 5.73i)T + 169iT^{2} \) |
| 17 | \( 1 - 23.3T + 289T^{2} \) |
| 19 | \( 1 + (11.7 - 11.7i)T - 361iT^{2} \) |
| 23 | \( 1 - 5.80T + 529T^{2} \) |
| 29 | \( 1 + (18.3 + 18.3i)T + 841iT^{2} \) |
| 31 | \( 1 - 16.9iT - 961T^{2} \) |
| 37 | \( 1 + (-15.3 + 15.3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 29.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (33.4 + 33.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 18.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-66.9 + 66.9i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (27.1 + 27.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-65.2 - 65.2i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-37.6 + 37.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 42.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 21.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-24.1 + 24.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 52.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 21.0T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04976109197476677090116839709, −9.755652665762294795933252643754, −8.742472483295503297634875923751, −7.925103110549683078590225450370, −6.81882771212165910798105164258, −5.62546739370187128104741457610, −5.00443690713875292027961537606, −3.92843697519877843091533639427, −2.07169258061870525796701193606, −1.22021903795105901477657515676,
1.35680123514872869661549488509, 2.61045428284715148001340726264, 3.74839930693501885865416675391, 5.17470996959424490623104621077, 6.11509110377582135670470602776, 6.81648557190474265421272948243, 7.88020895197566074396370643840, 8.986800413878124628833007833081, 9.701463214492395145063842912709, 10.67620289586406752740744171905