| L(s) = 1 | − 0.556·5-s + 9.52·7-s − 13.5·11-s − 10.5i·13-s − 2.63·17-s + (−0.112 − 18.9i)19-s − 22.0·23-s − 24.6·25-s + 48.7i·29-s + 0.248i·31-s − 5.30·35-s + 59.3i·37-s + 69.0i·41-s + 27.5·43-s + 44.8·47-s + ⋯ |
| L(s) = 1 | − 0.111·5-s + 1.36·7-s − 1.22·11-s − 0.810i·13-s − 0.155·17-s + (−0.00593 − 0.999i)19-s − 0.957·23-s − 0.987·25-s + 1.68i·29-s + 0.00800i·31-s − 0.151·35-s + 1.60i·37-s + 1.68i·41-s + 0.641·43-s + 0.955·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00593 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00593 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.331986711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331986711\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.112 + 18.9i)T \) |
| good | 5 | \( 1 + 0.556T + 25T^{2} \) |
| 7 | \( 1 - 9.52T + 49T^{2} \) |
| 11 | \( 1 + 13.5T + 121T^{2} \) |
| 13 | \( 1 + 10.5iT - 169T^{2} \) |
| 17 | \( 1 + 2.63T + 289T^{2} \) |
| 23 | \( 1 + 22.0T + 529T^{2} \) |
| 29 | \( 1 - 48.7iT - 841T^{2} \) |
| 31 | \( 1 - 0.248iT - 961T^{2} \) |
| 37 | \( 1 - 59.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 69.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 27.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 44.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 3.85iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 59.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 96.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 10.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 52.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 13.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 63.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 44.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 72.6iT - 9.40e3T^{2} \) |
| 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
−8.531725103822735173494331547136, −8.104073672991260167091194138478, −7.55617639753552557693492425578, −6.61616080083389137356538364427, −5.46992053527351011579810993120, −5.07086756345026601215250923184, −4.23651206214991550639468485994, −3.04293124755836111716094102146, −2.19670165142754109191733759039, −1.05755291209015067371543240391,
0.32115557435740177551521484377, 1.87284435911529408119054900398, 2.33716589467507955706415446447, 3.94922585746564822874755371547, 4.34365809347405337128871659230, 5.53098855333562361280396194447, 5.84574148348877766074637201687, 7.20811963952191910365545319885, 7.80087588832902603698821353346, 8.252953945097854970193684726891
