| L(s) = 1 | + (1.69 − 1.69i)5-s + 5.74·7-s + (5.59 + 5.59i)11-s + (13.5 + 13.5i)13-s − 19.7·17-s + (−21.6 + 21.6i)19-s + 24.9·23-s + 19.2i·25-s + (1.50 + 1.50i)29-s + 2.20i·31-s + (9.75 − 9.75i)35-s + (−27.6 + 27.6i)37-s − 51.3i·41-s + (21.4 + 21.4i)43-s − 76.5i·47-s + ⋯ |
| L(s) = 1 | + (0.339 − 0.339i)5-s + 0.820·7-s + (0.508 + 0.508i)11-s + (1.04 + 1.04i)13-s − 1.15·17-s + (−1.14 + 1.14i)19-s + 1.08·23-s + 0.768i·25-s + (0.0519 + 0.0519i)29-s + 0.0709i·31-s + (0.278 − 0.278i)35-s + (−0.748 + 0.748i)37-s − 1.25i·41-s + (0.498 + 0.498i)43-s − 1.62i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.193400906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.193400906\) |
| \(L(2)\) |
|
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 | \( 1 + (-1.69 + 1.69i)T - 25iT^{2} \) |
| 7 | \( 1 - 5.74T + 49T^{2} \) |
| 11 | \( 1 + (-5.59 - 5.59i)T + 121iT^{2} \) |
| 13 | \( 1 + (-13.5 - 13.5i)T + 169iT^{2} \) |
| 17 | \( 1 + 19.7T + 289T^{2} \) |
| 19 | \( 1 + (21.6 - 21.6i)T - 361iT^{2} \) |
| 23 | \( 1 - 24.9T + 529T^{2} \) |
| 29 | \( 1 + (-1.50 - 1.50i)T + 841iT^{2} \) |
| 31 | \( 1 - 2.20iT - 961T^{2} \) |
| 37 | \( 1 + (27.6 - 27.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-21.4 - 21.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 76.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (56.5 - 56.5i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-48.0 - 48.0i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-51.5 - 51.5i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-63.4 + 63.4i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 43.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 73.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 4.12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (38.4 - 38.4i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 52.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 23.1T + 9.40e3T^{2} \) |
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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
−9.595416320586724267539869890724, −8.775245263710057772618967383965, −8.414316259530539994409591129927, −7.09549631741662557130278893272, −6.48774960327278909418254404281, −5.43645166219747105564739660273, −4.48914750763648422179994551495, −3.78158201953847751815043127377, −2.07499097388912493587729900312, −1.38227493158715346966553792292,
0.68900672991576514952107912250, 2.04755688364247972250152027556, 3.12843049326683609110562185820, 4.30033152939215935474702379367, 5.16307136668455526112161770787, 6.27752530602615884705202794511, 6.75706607654223005736909353589, 8.046255909940862640424679945331, 8.601840518753123607136941454003, 9.331746144605593969033369023687
