| L(s) = 1 | + (0.947 − 1.05i)2-s + (−2.80 − 1.02i)3-s + (−0.206 − 1.98i)4-s + (−0.165 − 0.936i)5-s + (−3.72 + 1.97i)6-s + (2.67 + 1.54i)7-s + (−2.28 − 1.66i)8-s + (4.52 + 3.79i)9-s + (−1.13 − 0.713i)10-s + (3.39 − 1.95i)11-s + (−1.45 + 5.78i)12-s + (−0.284 − 0.781i)13-s + (4.15 − 1.34i)14-s + (−0.492 + 2.79i)15-s + (−3.91 + 0.820i)16-s + (−3.14 + 2.63i)17-s + ⋯ |
| L(s) = 1 | + (0.669 − 0.742i)2-s + (−1.61 − 0.589i)3-s + (−0.103 − 0.994i)4-s + (−0.0738 − 0.418i)5-s + (−1.52 + 0.807i)6-s + (1.01 + 0.583i)7-s + (−0.807 − 0.589i)8-s + (1.50 + 1.26i)9-s + (−0.360 − 0.225i)10-s + (1.02 − 0.590i)11-s + (−0.419 + 1.67i)12-s + (−0.0789 − 0.216i)13-s + (1.11 − 0.359i)14-s + (−0.127 + 0.721i)15-s + (−0.978 + 0.205i)16-s + (−0.762 + 0.639i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.514789 - 0.707049i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.514789 - 0.707049i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.947 + 1.05i)T \) |
| 19 | \( 1 + (-0.473 - 4.33i)T \) |
| good | 3 | \( 1 + (2.80 + 1.02i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.165 + 0.936i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.67 - 1.54i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.39 + 1.95i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.284 + 0.781i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (3.14 - 2.63i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (2.45 + 0.432i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.95 + 2.32i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.560 - 0.970i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 11.0iT - 37T^{2} \) |
| 41 | \( 1 + (-1.74 + 4.80i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.92 - 0.515i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.23 + 2.66i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (4.27 + 0.753i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.677 + 0.568i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.74 - 9.90i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.32 + 5.30i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.66 - 9.42i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.43 - 0.887i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-3.07 - 1.12i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (10.7 + 6.18i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.254 + 0.698i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (6.10 + 7.27i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−13.91471315652313117602297532632, −12.67301636547357557686326576500, −11.92555034479689417780181441555, −11.36787382210202428010413144313, −10.31622034048772232023882777325, −8.517854011509386315809875954009, −6.49602938983851306049231555094, −5.58085289736938256493778600545, −4.45529510890374650940717094800, −1.46619057234486424028843596408,
4.24369600601713487306901091477, 4.99949078248819031807519873528, 6.47505678974151119783452211927, 7.29271140227970893873436182689, 9.267594224583937575932464775677, 10.94867896721371091364041810706, 11.47515256552437138114319317565, 12.51439272528910896197752344741, 14.02182533454898642517933398387, 14.93941241228233406879184730216
