| L(s) = 1 | + (2.22 + 0.391i)2-s + (1.67 + 0.433i)3-s + (2.90 + 1.05i)4-s + (−3.20 − 1.16i)5-s + (3.55 + 1.61i)6-s + (−2.62 + 0.334i)7-s + (2.12 + 1.22i)8-s + (2.62 + 1.45i)9-s + (−6.66 − 3.84i)10-s + (0.870 + 2.39i)11-s + (4.40 + 3.02i)12-s + (2.11 − 5.80i)13-s + (−5.96 − 0.285i)14-s + (−4.87 − 3.34i)15-s + (−0.494 − 0.414i)16-s + (−1.19 + 2.07i)17-s + ⋯ |
| L(s) = 1 | + (1.57 + 0.276i)2-s + (0.968 + 0.250i)3-s + (1.45 + 0.527i)4-s + (−1.43 − 0.522i)5-s + (1.45 + 0.661i)6-s + (−0.991 + 0.126i)7-s + (0.750 + 0.433i)8-s + (0.874 + 0.484i)9-s + (−2.10 − 1.21i)10-s + (0.262 + 0.720i)11-s + (1.27 + 0.874i)12-s + (0.585 − 1.60i)13-s + (−1.59 − 0.0763i)14-s + (−1.25 − 0.864i)15-s + (−0.123 − 0.103i)16-s + (−0.290 + 0.503i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.55783 + 0.524214i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55783 + 0.524214i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.67 - 0.433i)T \) |
| 7 | \( 1 + (2.62 - 0.334i)T \) |
| good | 2 | \( 1 + (-2.22 - 0.391i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (3.20 + 1.16i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.870 - 2.39i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.11 + 5.80i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.19 - 2.07i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.07 - 0.622i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.21 - 0.390i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.685 - 1.88i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.538 - 1.48i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + (-5.39 - 1.96i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.110 - 0.625i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (11.3 - 4.12i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (4.15 - 2.39i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.11 + 1.77i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.52 - 6.92i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.39 + 7.89i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.45 + 2.57i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 + (-0.167 + 0.951i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.38 + 0.867i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (4.97 + 8.61i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.93 + 1.39i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.77168205035570000973455545831, −12.29727394790400018262807951571, −10.93799366700131045804041861864, −9.583328742640792634511698580881, −8.288862932537377365114160399964, −7.47727046330534327650026038938, −6.18560187481557300363905984325, −4.68124081432084490372662373765, −3.82357006086231763011137390437, −3.02565122958241686463386195009,
2.66629672583218190186449802118, 3.75606178219153417659189006003, 4.19837528805628302657160926213, 6.36535618377997271239944971363, 6.97751019537717573801746619598, 8.327278833725683922404069364883, 9.492257644196281852843400005810, 11.16646343340201348212605652052, 11.68646114944430281793682631274, 12.70359520173936830664116652693
