| L(s) = 1 | + (−0.207 + 1.39i)2-s + (−1.63 + 0.562i)3-s + (−1.91 − 0.581i)4-s + (−0.342 + 0.939i)5-s + (−0.446 − 2.40i)6-s + (3.58 + 0.632i)7-s + (1.21 − 2.55i)8-s + (2.36 − 1.84i)9-s + (−1.24 − 0.673i)10-s + (5.64 − 2.05i)11-s + (3.46 − 0.123i)12-s + (−3.21 + 2.69i)13-s + (−1.62 + 4.88i)14-s + (0.0317 − 1.73i)15-s + (3.32 + 2.22i)16-s + (−1.97 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.147 + 0.989i)2-s + (−0.945 + 0.324i)3-s + (−0.956 − 0.290i)4-s + (−0.152 + 0.420i)5-s + (−0.182 − 0.983i)6-s + (1.35 + 0.238i)7-s + (0.428 − 0.903i)8-s + (0.789 − 0.614i)9-s + (−0.393 − 0.213i)10-s + (1.70 − 0.619i)11-s + (0.999 − 0.0356i)12-s + (−0.890 + 0.747i)13-s + (−0.435 + 1.30i)14-s + (0.00821 − 0.447i)15-s + (0.830 + 0.556i)16-s + (−0.477 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.557469 + 0.902696i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.557469 + 0.902696i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.207 - 1.39i)T \) |
| 3 | \( 1 + (1.63 - 0.562i)T \) |
| 5 | \( 1 + (0.342 - 0.939i)T \) |
| good | 7 | \( 1 + (-3.58 - 0.632i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-5.64 + 2.05i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.21 - 2.69i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.97 - 1.13i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.70 - 3.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.02 + 5.79i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.26 + 1.50i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (7.49 - 1.32i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.70 - 4.68i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.78 - 3.32i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.67 - 4.59i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.129 + 0.737i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 1.52iT - 53T^{2} \) |
| 59 | \( 1 + (-3.24 - 1.18i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.266 + 1.51i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.49 - 10.1i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.22 - 2.12i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.32 - 7.48i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.99 + 3.57i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.175 - 0.147i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (8.63 + 4.98i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.147 + 0.0535i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.30726322451791363520977558626, −10.07997395126282811207048419187, −9.300383919038114802346105023146, −8.368062639152493985282666636379, −7.30549929912129739061107876326, −6.53256884320493369669074797086, −5.66960157836294936584522377104, −4.66688000839193502402063637565, −3.92555809358133059799577391680, −1.33432832551390613433470301665,
0.931031713799588702608836904966, 1.94146963341591016495521616874, 3.89515234392423938274465926306, 4.85110941409045127118434535240, 5.43189055548132745493888637429, 7.23139194348149213030920812504, 7.70142519241758427606957977117, 9.103129527967827977511809473468, 9.680513628435023187665682167762, 10.90310069014166996134556207301
