| L(s) = 1 | + (1.05 − 0.943i)2-s + (0.603 + 1.62i)3-s + (0.221 − 1.98i)4-s + (−2.74 + 1.92i)5-s + (2.16 + 1.14i)6-s + (−3.40 + 2.85i)7-s + (−1.64 − 2.30i)8-s + (−2.27 + 1.95i)9-s + (−1.07 + 4.61i)10-s + (1.34 + 0.940i)11-s + (3.36 − 0.839i)12-s + (0.598 + 0.279i)13-s + (−0.893 + 6.22i)14-s + (−4.77 − 3.29i)15-s + (−3.90 − 0.878i)16-s + (6.26 + 3.61i)17-s + ⋯ |
| L(s) = 1 | + (0.745 − 0.666i)2-s + (0.348 + 0.937i)3-s + (0.110 − 0.993i)4-s + (−1.22 + 0.858i)5-s + (0.884 + 0.466i)6-s + (−1.28 + 1.08i)7-s + (−0.580 − 0.814i)8-s + (−0.757 + 0.652i)9-s + (−0.341 + 1.45i)10-s + (0.405 + 0.283i)11-s + (0.970 − 0.242i)12-s + (0.166 + 0.0774i)13-s + (−0.238 + 1.66i)14-s + (−1.23 − 0.850i)15-s + (−0.975 − 0.219i)16-s + (1.52 + 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0743 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0743 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.896407 + 0.965696i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.896407 + 0.965696i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.05 + 0.943i)T \) |
| 3 | \( 1 + (-0.603 - 1.62i)T \) |
| good | 5 | \( 1 + (2.74 - 1.92i)T + (1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (3.40 - 2.85i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-1.34 - 0.940i)T + (3.76 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-0.598 - 0.279i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-6.26 - 3.61i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.158 + 0.0425i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (1.13 - 1.34i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.579 - 0.270i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-3.95 + 4.71i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.68 + 6.27i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.09 + 0.761i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (6.12 - 8.75i)T + (-14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (1.66 - 1.39i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-9.05 - 9.05i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.76 - 8.23i)T + (-20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (7.14 + 0.624i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (3.41 - 7.31i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-10.5 - 6.09i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-14.1 + 8.17i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.128 - 0.352i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (2.11 + 4.54i)T + (-53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (1.44 + 2.50i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.447 + 2.53i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.53140495974044191461157298251, −10.49166142941333953247460281768, −9.842196121542971950155370481027, −8.998525021621391938775823824646, −7.75428334497436732237174041915, −6.38943438464965142622718592048, −5.57438603981876129249985445331, −4.07057830687486544022649949005, −3.46623582491945950382401829017, −2.64873824830509068440909534352,
0.63067497296516725622223344774, 3.26701360456591530913511226232, 3.75976362473741023395364328401, 5.14701061635223907766330353989, 6.49194559305830842623735474579, 7.12548172798921744114786004519, 7.958932488337428320716807026779, 8.625392906055959650359765941404, 9.834354986408995580429374094049, 11.43626851853596828346683300691
