| L(s) = 1 | + 8.89i·5-s + 2.82i·7-s − 18.2·11-s − 5.79i·13-s + 21.5·17-s − 18.2·19-s + 33.3i·23-s − 54.1·25-s + 4.49i·29-s + 2.25i·31-s − 25.1·35-s − 43.1i·37-s + 1.59·41-s − 63.4·43-s − 72.3i·47-s + ⋯ |
| L(s) = 1 | + 1.77i·5-s + 0.404i·7-s − 1.65·11-s − 0.445i·13-s + 1.27·17-s − 0.960·19-s + 1.45i·23-s − 2.16·25-s + 0.154i·29-s + 0.0728i·31-s − 0.719·35-s − 1.16i·37-s + 0.0389·41-s − 1.47·43-s − 1.54i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3657984314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3657984314\) |
| \(L(2)\) |
|
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 | \( 1 - 8.89iT - 25T^{2} \) |
| 7 | \( 1 - 2.82iT - 49T^{2} \) |
| 11 | \( 1 + 18.2T + 121T^{2} \) |
| 13 | \( 1 + 5.79iT - 169T^{2} \) |
| 17 | \( 1 - 21.5T + 289T^{2} \) |
| 19 | \( 1 + 18.2T + 361T^{2} \) |
| 23 | \( 1 - 33.3iT - 529T^{2} \) |
| 29 | \( 1 - 4.49iT - 841T^{2} \) |
| 31 | \( 1 - 2.25iT - 961T^{2} \) |
| 37 | \( 1 + 43.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.59T + 1.68e3T^{2} \) |
| 43 | \( 1 + 63.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 70.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 63.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 3.24T + 4.48e3T^{2} \) |
| 71 | \( 1 + 68.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 35.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 42.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 5.19T + 7.92e3T^{2} \) |
| 97 | \( 1 + 26.8T + 9.40e3T^{2} \) |
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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
−10.39572510101386924227797795452, −9.533195748560749811013449266733, −8.211228858263383503257241527688, −7.61544923228234786680452979431, −6.91838980199005977672378487566, −5.82618230413719589309373378992, −5.30862362766208351459119822695, −3.66252682945640314197790963145, −2.95207141629920803379418016012, −2.10078972520905259451901422199,
0.11077248205690971206278527659, 1.24023446148577951777020393494, 2.55560909186691030043161322201, 4.02508327626081786694368661513, 4.85471708336649194397752136098, 5.40394011579210332316734476850, 6.50404039435229067612922487588, 7.80908123989956716624887965418, 8.226627547038395466104963457929, 8.950693821847173926737581760307
