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
−9.790753913266241288793745958826, −8.833391264601046049648730710372, −7.65726354718871729799489191212, −7.23383721315368271444815770320, −6.42012430524539809875510484842, −5.42845865419309550388750751169, −4.20470712469692099644450984888, −3.08977708324495851708654646073, −2.78915727400149060813222056347, −0.43130828294543114500710861957,
0.54292822189514017021420904768, 2.27986404076872402641575823610, 3.52507559968836338808885153539, 4.35957761836568765736502906313, 5.15533074711540325000937991897, 6.44169470118386819940595994946, 7.07516310233099046496856374769, 7.965148124322713339475663205205, 8.935850101075332083409327861116, 9.513195352125931145582462433164