| L(s) = 1 | + (−0.111 + 0.157i)2-s + (0.235 − 0.971i)3-s + (0.641 + 1.85i)4-s + (−0.171 + 3.59i)5-s + (0.126 + 0.145i)6-s + (−2.30 − 1.29i)7-s + (−0.733 − 0.215i)8-s + (−0.888 − 0.458i)9-s + (−0.545 − 0.428i)10-s + (1.58 + 2.21i)11-s + (1.95 − 0.186i)12-s + (0.00476 − 0.0331i)13-s + (0.461 − 0.218i)14-s + (3.45 + 1.01i)15-s + (−2.96 + 2.33i)16-s + (−7.18 + 1.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.0791 + 0.111i)2-s + (0.136 − 0.561i)3-s + (0.320 + 0.927i)4-s + (−0.0765 + 1.60i)5-s + (0.0515 + 0.0595i)6-s + (−0.872 − 0.488i)7-s + (−0.259 − 0.0761i)8-s + (−0.296 − 0.152i)9-s + (−0.172 − 0.135i)10-s + (0.476 + 0.669i)11-s + (0.564 − 0.0538i)12-s + (0.00132 − 0.00919i)13-s + (0.123 − 0.0583i)14-s + (0.890 + 0.261i)15-s + (−0.742 + 0.583i)16-s + (−1.74 + 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.516610 + 0.946838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.516610 + 0.946838i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (2.30 + 1.29i)T \) |
| 23 | \( 1 + (-1.86 + 4.41i)T \) |
| good | 2 | \( 1 + (0.111 - 0.157i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (0.171 - 3.59i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (-1.58 - 2.21i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-0.00476 + 0.0331i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (7.18 - 1.38i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-1.20 - 0.231i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (-6.71 - 7.74i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (2.29 - 2.18i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (8.07 + 4.16i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-8.26 - 5.30i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.84 + 0.541i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-4.78 - 8.28i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.54 - 1.82i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (0.227 + 0.179i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (2.27 + 9.39i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (-13.4 - 1.28i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (-1.59 - 3.49i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-3.27 - 9.47i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (-14.8 + 5.94i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (3.51 - 2.26i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (1.14 + 1.09i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (10.8 + 6.95i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.03630954751839408686568358758, −10.74627616452968884749917970997, −9.436334616281568583020418515276, −8.469549492691309111271898811755, −7.21341205309873354289786733965, −6.88560690715923542300347673454, −6.36997401251352308126567083883, −4.19798052329955884433369920666, −3.17887534926606868069081473918, −2.36184124511924427967571814071,
0.63254619755437406314836604168, 2.32908888111339424353297567291, 3.95617606027818213337378617689, 5.03966273047272429448104606040, 5.78641645151551856263361643234, 6.77140533781431440397429098862, 8.416336282667658890081766725544, 9.189951033203268090360174385306, 9.449961669190406242698821399465, 10.60725919845877400911934949283
