| L(s) = 1 | + (0.915 − 2.00i)2-s + (0.959 + 0.281i)3-s + (−1.87 − 2.15i)4-s + (−2.61 + 1.68i)5-s + (1.44 − 1.66i)6-s + (−0.427 + 2.97i)7-s + (−1.80 + 0.531i)8-s + (0.841 + 0.540i)9-s + (0.976 + 6.79i)10-s + (−2.48 − 5.43i)11-s + (−1.18 − 2.59i)12-s + (−0.0566 − 0.393i)13-s + (5.57 + 3.58i)14-s + (−2.98 + 0.877i)15-s + (0.221 − 1.53i)16-s + (0.862 − 0.995i)17-s + ⋯ |
| L(s) = 1 | + (0.647 − 1.41i)2-s + (0.553 + 0.162i)3-s + (−0.935 − 1.07i)4-s + (−1.17 + 0.752i)5-s + (0.589 − 0.679i)6-s + (−0.161 + 1.12i)7-s + (−0.639 + 0.187i)8-s + (0.280 + 0.180i)9-s + (0.308 + 2.14i)10-s + (−0.748 − 1.63i)11-s + (−0.342 − 0.749i)12-s + (−0.0157 − 0.109i)13-s + (1.48 + 0.957i)14-s + (−0.771 + 0.226i)15-s + (0.0553 − 0.384i)16-s + (0.209 − 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.945399 - 0.717563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.945399 - 0.717563i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-2.43 - 4.13i)T \) |
| good | 2 | \( 1 + (-0.915 + 2.00i)T + (-1.30 - 1.51i)T^{2} \) |
| 5 | \( 1 + (2.61 - 1.68i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.427 - 2.97i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (2.48 + 5.43i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.0566 + 0.393i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.862 + 0.995i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.67 - 1.93i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (0.836 - 0.965i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.575 + 0.169i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (6.75 + 4.34i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (7.15 - 4.59i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-7.26 - 2.13i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 + (-0.399 + 2.78i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.477 - 3.32i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.182 + 0.0536i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.73 + 8.18i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (5.61 - 12.3i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (9.37 + 10.8i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.718 + 4.99i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-6.86 - 4.41i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (0.416 + 0.122i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-0.265 + 0.170i)T + (40.2 - 88.2i)T^{2} \) |
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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
−14.28814307254460397590581478039, −13.30921091504589360407192390936, −12.11067926355913698118373085325, −11.34793909614047840511799135882, −10.47781032179285153928788860904, −8.972629295530091978883143878345, −7.68300750258443462462951847357, −5.46312092844551907230211753743, −3.57112208244166780267811491598, −2.85783205743221049117445864961,
4.03423105257628206892948813824, 4.88740516858702272645932748937, 7.07268702928316015884821927294, 7.52921500319124319788946549231, 8.640380827891177972968687855733, 10.35055630441435860270506728818, 12.27460053660880516049586122447, 13.06644580798020493499830101159, 14.06258906747576780174392316339, 15.21997630880887204884130058526
