| L(s) = 1 | − 4.24·5-s − 8.48·7-s + 4·11-s + 18i·13-s − 4.24i·17-s − 16.9i·19-s + 36i·23-s − 7.00·25-s − 12.7·29-s + 8.48·31-s + 35.9·35-s − 36i·37-s + 29.6i·41-s − 67.8i·43-s − 36i·47-s + ⋯ |
| L(s) = 1 | − 0.848·5-s − 1.21·7-s + 0.363·11-s + 1.38i·13-s − 0.249i·17-s − 0.893i·19-s + 1.56i·23-s − 0.280·25-s − 0.438·29-s + 0.273·31-s + 1.02·35-s − 0.972i·37-s + 0.724i·41-s − 1.57i·43-s − 0.765i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8904762063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8904762063\) |
| \(L(2)\) |
|
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 | \( 1 + 4.24T + 25T^{2} \) |
| 7 | \( 1 + 8.48T + 49T^{2} \) |
| 11 | \( 1 - 4T + 121T^{2} \) |
| 13 | \( 1 - 18iT - 169T^{2} \) |
| 17 | \( 1 + 4.24iT - 289T^{2} \) |
| 19 | \( 1 + 16.9iT - 361T^{2} \) |
| 23 | \( 1 - 36iT - 529T^{2} \) |
| 29 | \( 1 + 12.7T + 841T^{2} \) |
| 31 | \( 1 - 8.48T + 961T^{2} \) |
| 37 | \( 1 + 36iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 29.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 67.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 36iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 80T + 3.48e3T^{2} \) |
| 61 | \( 1 + 36iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 118. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 108iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 56T + 5.32e3T^{2} \) |
| 79 | \( 1 - 25.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 76T + 6.88e3T^{2} \) |
| 89 | \( 1 - 89.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 104T + 9.40e3T^{2} \) |
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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
−9.298921175319995237299565560248, −8.950763580425735391337378219746, −7.64808501716893239606993233020, −7.01463246182624451805874114611, −6.30835572071227202635511446185, −5.16284948128031563052929117711, −3.97168226338377604423732315746, −3.47449546861867262173630192698, −2.08341261330756040288038127505, −0.38130728394347424710187129916,
0.805207367644638355913491045223, 2.66820193789950491497642490374, 3.53726560943620849882192016433, 4.31957144254675947273876415785, 5.64729178007761870323325434888, 6.36401765663643260902450700004, 7.28417414607325586301958567222, 8.135159479315443331983131105751, 8.765653173953871542116874887869, 10.01409027489600292213011761303
