L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)3-s + (−0.866 − 0.499i)4-s + (0.0678 + 0.0181i)5-s + (−0.707 + 0.707i)6-s + (0.873 − 2.49i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.0351 − 0.0608i)10-s + (0.152 + 0.570i)11-s + (0.5 + 0.866i)12-s + (−1.58 − 3.24i)13-s + (−2.18 − 1.48i)14-s + (−0.0496 − 0.0496i)15-s + (0.500 + 0.866i)16-s + (2.62 − 4.54i)17-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.499 − 0.288i)3-s + (−0.433 − 0.249i)4-s + (0.0303 + 0.00813i)5-s + (−0.288 + 0.288i)6-s + (0.330 − 0.943i)7-s + (−0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (0.0111 − 0.0192i)10-s + (0.0460 + 0.171i)11-s + (0.144 + 0.249i)12-s + (−0.438 − 0.898i)13-s + (−0.584 − 0.398i)14-s + (−0.0128 − 0.0128i)15-s + (0.125 + 0.216i)16-s + (0.635 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.101971 - 0.957053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.101971 - 0.957053i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.873 + 2.49i)T \) |
| 13 | \( 1 + (1.58 + 3.24i)T \) |
good | 5 | \( 1 + (-0.0678 - 0.0181i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.152 - 0.570i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.62 + 4.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.64 + 1.51i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (4.01 - 2.31i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 + (-2.20 - 8.24i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (6.86 + 1.83i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.36 + 3.36i)T - 41iT^{2} \) |
| 43 | \( 1 - 1.42iT - 43T^{2} \) |
| 47 | \( 1 + (-2.91 + 10.8i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.67 + 2.89i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.22 + 1.13i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.23 + 0.713i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.82 - 2.63i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-8.04 - 8.04i)T + 71iT^{2} \) |
| 73 | \( 1 + (-11.0 + 2.94i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.725 + 1.25i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.15 + 7.15i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.33 - 16.1i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-9.94 + 9.94i)T - 97iT^{2} \) |
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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
−10.32321030514597003227086074212, −10.04023068685873666927050304845, −8.623234038987620567757662135837, −7.64390327016820213081271643499, −6.79091160086454085926087010170, −5.52105325570248079586264129831, −4.68350382866554852817087407763, −3.56365845933613374635111648170, −2.09934123148114947491995886019, −0.54931672822282157515119230184,
2.08146379650024786433327251186, 3.87172744023677608036972648455, 4.73348869231812350113379640375, 5.95250910329127010455850827584, 6.25365222653490085111318042103, 7.68698422583206785765938153084, 8.470625700589684304275231555146, 9.364612076286617042952974037181, 10.25034950470957432237924712121, 11.30832564392977539175242552817