| L(s) = 1 | + (−0.529 + 0.916i)2-s + (1.30 + 1.09i)3-s + (0.440 + 0.762i)4-s + (0.109 − 0.621i)5-s + (−1.69 + 0.618i)6-s + (0.171 − 0.974i)7-s − 3.04·8-s + (−0.0143 − 0.0812i)9-s + (0.511 + 0.429i)10-s + (−0.0872 + 0.0731i)11-s + (−0.261 + 1.48i)12-s + (−0.0924 + 0.524i)13-s + (0.802 + 0.673i)14-s + (0.826 − 0.693i)15-s + (0.732 − 1.26i)16-s + (−1.31 + 2.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.374 + 0.647i)2-s + (0.755 + 0.633i)3-s + (0.220 + 0.381i)4-s + (0.0490 − 0.278i)5-s + (−0.693 + 0.252i)6-s + (0.0649 − 0.368i)7-s − 1.07·8-s + (−0.00477 − 0.0270i)9-s + (0.161 + 0.135i)10-s + (−0.0262 + 0.0220i)11-s + (−0.0753 + 0.427i)12-s + (−0.0256 + 0.145i)13-s + (0.214 + 0.179i)14-s + (0.213 − 0.179i)15-s + (0.183 − 0.317i)16-s + (−0.317 + 0.550i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.831924 + 0.711330i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.831924 + 0.711330i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-3.35 + 9.88i)T \) |
| good | 2 | \( 1 + (0.529 - 0.916i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.30 - 1.09i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.109 + 0.621i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.171 + 0.974i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.0872 - 0.0731i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.0924 - 0.524i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.31 - 2.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 3.35i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.67 + 4.63i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.85 + 1.55i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.484 - 2.74i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (0.424 + 2.40i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 - 0.370T + 41T^{2} \) |
| 43 | \( 1 + (3.71 - 6.43i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.25 + 0.821i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.12 - 12.0i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (4.19 - 3.51i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (5.62 - 2.04i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.712 + 4.03i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (1.20 + 2.09i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (10.9 - 9.18i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (1.79 + 10.2i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (5.58 + 4.68i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-10.4 + 3.80i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.77 + 10.0i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−14.23360638585748980247193637524, −12.97310663140070669145004968761, −11.88839323133410011992211797894, −10.54216744870315816843185141422, −9.233801761559056582085679818585, −8.621789296978600036691059791431, −7.47834322661074632543833570357, −6.26360398758844371268222843020, −4.39054294269358166420309356702, −2.97940536820223506150269508111,
1.86036123680356588892849743215, 3.09659163532995201505500018997, 5.48339716117799928201488429714, 6.89723605426703922465834259629, 8.135602127181244911991379941579, 9.228375805526715392969991026667, 10.26323492780352373853553638597, 11.35381971656724243848156627072, 12.28499465298168980154865709478, 13.48515492343571571971199583842
