L(s) = 1 | + (0.369 + 0.0990i)2-s − 0.914i·3-s + (−1.60 − 0.926i)4-s + (3.58 − 0.959i)5-s + (0.0906 − 0.338i)6-s + (−2.28 + 1.34i)7-s + (−1.04 − 1.04i)8-s + 2.16·9-s + 1.41·10-s + (0.0619 + 0.0619i)11-s + (−0.847 + 1.46i)12-s + (−1.63 + 3.21i)13-s + (−0.975 + 0.270i)14-s + (−0.877 − 3.27i)15-s + (1.57 + 2.72i)16-s + (−2.94 + 5.10i)17-s + ⋯ |
L(s) = 1 | + (0.261 + 0.0700i)2-s − 0.528i·3-s + (−0.802 − 0.463i)4-s + (1.60 − 0.429i)5-s + (0.0369 − 0.138i)6-s + (−0.861 + 0.507i)7-s + (−0.368 − 0.368i)8-s + 0.721·9-s + 0.448·10-s + (0.0186 + 0.0186i)11-s + (−0.244 + 0.423i)12-s + (−0.453 + 0.891i)13-s + (−0.260 + 0.0722i)14-s + (−0.226 − 0.845i)15-s + (0.392 + 0.680i)16-s + (−0.714 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04766 - 0.340228i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04766 - 0.340228i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.28 - 1.34i)T \) |
| 13 | \( 1 + (1.63 - 3.21i)T \) |
good | 2 | \( 1 + (-0.369 - 0.0990i)T + (1.73 + i)T^{2} \) |
| 3 | \( 1 + 0.914iT - 3T^{2} \) |
| 5 | \( 1 + (-3.58 + 0.959i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.0619 - 0.0619i)T + 11iT^{2} \) |
| 17 | \( 1 + (2.94 - 5.10i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.62 + 2.62i)T + 19iT^{2} \) |
| 23 | \( 1 + (0.386 - 0.223i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.706 + 1.22i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.520 - 1.94i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-0.686 + 2.56i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (3.00 - 0.804i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-8.64 + 4.99i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.36 - 8.84i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.28 + 10.8i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.85 + 6.90i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 0.127iT - 61T^{2} \) |
| 67 | \( 1 + (7.04 - 7.04i)T - 67iT^{2} \) |
| 71 | \( 1 + (9.83 + 2.63i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (8.84 + 2.37i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.75 + 3.04i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.17 - 2.17i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.63 - 0.437i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.452 + 1.68i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−13.70741558522952168895160865031, −13.02174225161068824761437547853, −12.51060639390070675995567495491, −10.39581952970587305361747368711, −9.504695837927197885159772079272, −8.834838521641291544217829200161, −6.63328766379353507754416523951, −5.91315764701906469204554986821, −4.49070560909224291375597750264, −1.93825494747935502300175402542,
2.92867562524603254954866562350, 4.52085617507822355613955151400, 5.86017203689688303490661503812, 7.25030048551033902149471700457, 9.123901800208813586117681737096, 9.842756860967080347860951633289, 10.54852415875092201804875211143, 12.50136875448745289813570195079, 13.30507409457437956830610968551, 13.90200142892559881723901744186