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
−11.30832564392977539175242552817, −10.25034950470957432237924712121, −9.364612076286617042952974037181, −8.470625700589684304275231555146, −7.68698422583206785765938153084, −6.25365222653490085111318042103, −5.95250910329127010455850827584, −4.73348869231812350113379640375, −3.87172744023677608036972648455, −2.08146379650024786433327251186,
0.54931672822282157515119230184, 2.09934123148114947491995886019, 3.56365845933613374635111648170, 4.68350382866554852817087407763, 5.52105325570248079586264129831, 6.79091160086454085926087010170, 7.64390327016820213081271643499, 8.623234038987620567757662135837, 10.04023068685873666927050304845, 10.32321030514597003227086074212