| L(s) = 1 | + (−1.18 + 0.732i)2-s + (−0.0275 + 0.0552i)4-s + (2.15 − 0.835i)5-s + (−0.227 + 2.45i)7-s + (−0.264 − 2.85i)8-s + (−1.94 + 2.57i)10-s + (−0.646 + 0.400i)11-s + (−0.169 + 1.82i)13-s + (−1.52 − 3.06i)14-s + (2.33 + 3.09i)16-s + (−4.73 + 4.31i)17-s + (1.10 − 3.88i)19-s + (−0.0131 + 0.142i)20-s + (0.472 − 0.948i)22-s + (1.32 + 0.817i)23-s + ⋯ |
| L(s) = 1 | + (−0.836 + 0.518i)2-s + (−0.0137 + 0.0276i)4-s + (0.965 − 0.373i)5-s + (−0.0858 + 0.926i)7-s + (−0.0936 − 1.01i)8-s + (−0.613 + 0.813i)10-s + (−0.195 + 0.120i)11-s + (−0.0469 + 0.506i)13-s + (−0.408 − 0.819i)14-s + (0.583 + 0.772i)16-s + (−1.14 + 1.04i)17-s + (0.253 − 0.891i)19-s + (−0.00294 + 0.0318i)20-s + (0.100 − 0.202i)22-s + (0.275 + 0.170i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.266300 + 0.764301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266300 + 0.764301i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 + (6.52 + 7.77i)T \) |
| good | 2 | \( 1 + (1.18 - 0.732i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (-2.15 + 0.835i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (0.227 - 2.45i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (0.646 - 0.400i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (0.169 - 1.82i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (4.73 - 4.31i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.10 + 3.88i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (-1.32 - 0.817i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (3.77 - 1.46i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-5.80 - 7.68i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (1.84 + 0.344i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (-5.85 - 2.27i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (0.0530 - 0.00991i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 6.18T + 47T^{2} \) |
| 53 | \( 1 + (1.57 - 5.53i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (0.0292 + 0.315i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (1.10 - 1.00i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.265 - 2.86i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (10.6 + 4.13i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (-5.13 - 1.98i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (8.01 - 3.10i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (1.11 - 11.9i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (-1.77 - 3.57i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (10.0 + 9.18i)T + (8.95 + 96.5i)T^{2} \) |
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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
−10.06812698118576645007332699688, −9.156506795779938329625113520649, −8.983692298671681421186616577890, −8.114885517776513005942123579210, −6.97074339609024617607763822598, −6.31329121856351883533953267905, −5.37344566162024277933081684130, −4.29714634005558417440111082215, −2.81017111823667519167920549793, −1.56925946557848556330001993818,
0.50466400856089461292145075160, 1.91243373744505072927073705076, 2.87051364179783520696561321730, 4.35851068646296124909664145638, 5.47560755399069215654436541402, 6.29535969808055154617414497040, 7.37011937263972195410591321918, 8.194829705749051922622054553558, 9.244998912503468167527657624826, 9.843093637390193615059622747223
