| L(s) = 1 | − 7.65·5-s − 1.65i·7-s − 1.51i·11-s − 0.343·13-s + 13.3·17-s − 20.8i·19-s − 33.6i·23-s + 33.6·25-s − 39.6·29-s + 45.2i·31-s + 12.6i·35-s − 29.5·37-s − 24.6·41-s + 50.0i·43-s + 35.3i·47-s + ⋯ |
| L(s) = 1 | − 1.53·5-s − 0.236i·7-s − 0.137i·11-s − 0.0263·13-s + 0.783·17-s − 1.09i·19-s − 1.46i·23-s + 1.34·25-s − 1.36·29-s + 1.45i·31-s + 0.362i·35-s − 0.799·37-s − 0.600·41-s + 1.16i·43-s + 0.751i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6727244785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6727244785\) |
| \(L(2)\) |
|
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 | \( 1 + 7.65T + 25T^{2} \) |
| 7 | \( 1 + 1.65iT - 49T^{2} \) |
| 11 | \( 1 + 1.51iT - 121T^{2} \) |
| 13 | \( 1 + 0.343T + 169T^{2} \) |
| 17 | \( 1 - 13.3T + 289T^{2} \) |
| 19 | \( 1 + 20.8iT - 361T^{2} \) |
| 23 | \( 1 + 33.6iT - 529T^{2} \) |
| 29 | \( 1 + 39.6T + 841T^{2} \) |
| 31 | \( 1 - 45.2iT - 961T^{2} \) |
| 37 | \( 1 + 29.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 24.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 53.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 62.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 55.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 137. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 114. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 2.56T + 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−9.842848196813900977455361782734, −8.743791846838004308082255443057, −8.228780380180965450475044443255, −7.28487529070865541532569351479, −6.79237435667556819056643929215, −5.42646902145214679156303031616, −4.48947330184999427874938858133, −3.70344118577338529385707351447, −2.75492701746988251290542984514, −0.953225677013276026226056673200,
0.25851254791083164114166383014, 1.85961928747459417730076731839, 3.52941693951995641179996348667, 3.80158161839268499609474282273, 5.11645030300561293873492710694, 5.92778155714145176842298697262, 7.27938063421854097302403492524, 7.65541137758784623336373704128, 8.427815636255374818633159845532, 9.369047245555417241909831668774
