L(s) = 1 | + (−4.78 − 4.78i)5-s − 10.3·7-s + (−0.526 + 0.526i)11-s + (−17.2 + 17.2i)13-s − 4.71·17-s + (2.53 + 2.53i)19-s + 12.5·23-s + 20.8i·25-s + (−2.19 + 2.19i)29-s − 28.0i·31-s + (49.4 + 49.4i)35-s + (32.1 + 32.1i)37-s − 23.1i·41-s + (−4.79 + 4.79i)43-s − 39.0i·47-s + ⋯ |
L(s) = 1 | + (−0.957 − 0.957i)5-s − 1.47·7-s + (−0.0478 + 0.0478i)11-s + (−1.32 + 1.32i)13-s − 0.277·17-s + (0.133 + 0.133i)19-s + 0.547·23-s + 0.834i·25-s + (−0.0757 + 0.0757i)29-s − 0.904i·31-s + (1.41 + 1.41i)35-s + (0.867 + 0.867i)37-s − 0.563i·41-s + (−0.111 + 0.111i)43-s − 0.829i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7225329894\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7225329894\) |
\(L(2)\) |
|
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 | \( 1 + (4.78 + 4.78i)T + 25iT^{2} \) |
| 7 | \( 1 + 10.3T + 49T^{2} \) |
| 11 | \( 1 + (0.526 - 0.526i)T - 121iT^{2} \) |
| 13 | \( 1 + (17.2 - 17.2i)T - 169iT^{2} \) |
| 17 | \( 1 + 4.71T + 289T^{2} \) |
| 19 | \( 1 + (-2.53 - 2.53i)T + 361iT^{2} \) |
| 23 | \( 1 - 12.5T + 529T^{2} \) |
| 29 | \( 1 + (2.19 - 2.19i)T - 841iT^{2} \) |
| 31 | \( 1 + 28.0iT - 961T^{2} \) |
| 37 | \( 1 + (-32.1 - 32.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 23.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (4.79 - 4.79i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 39.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (27.9 + 27.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-79.8 + 79.8i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-36.7 + 36.7i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-10.9 - 10.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 52.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 67.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 56.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (58.3 + 58.3i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 60.9T + 9.40e3T^{2} \) |
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
−9.513195352125931145582462433164, −8.935850101075332083409327861116, −7.965148124322713339475663205205, −7.07516310233099046496856374769, −6.44169470118386819940595994946, −5.15533074711540325000937991897, −4.35957761836568765736502906313, −3.52507559968836338808885153539, −2.27986404076872402641575823610, −0.54292822189514017021420904768,
0.43130828294543114500710861957, 2.78915727400149060813222056347, 3.08977708324495851708654646073, 4.20470712469692099644450984888, 5.42845865419309550388750751169, 6.42012430524539809875510484842, 7.23383721315368271444815770320, 7.65726354718871729799489191212, 8.833391264601046049648730710372, 9.790753913266241288793745958826