| L(s) = 1 | + (0.173 − 0.984i)2-s + (−1.43 − 1.20i)3-s + (−0.939 − 0.342i)4-s + (−1.87 + 0.684i)5-s + (−1.43 + 1.20i)6-s + (−2.53 − 4.38i)7-s + (−0.5 + 0.866i)8-s + (0.0923 + 0.524i)9-s + (0.347 + 1.96i)10-s + (0.705 − 1.22i)11-s + (0.939 + 1.62i)12-s + (−1 + 0.839i)13-s + (−4.75 + 1.73i)14-s + (3.53 + 1.28i)15-s + (0.766 + 0.642i)16-s + (0.414 − 2.35i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.831 − 0.697i)3-s + (−0.469 − 0.171i)4-s + (−0.840 + 0.305i)5-s + (−0.587 + 0.493i)6-s + (−0.957 − 1.65i)7-s + (−0.176 + 0.306i)8-s + (0.0307 + 0.174i)9-s + (0.109 + 0.622i)10-s + (0.212 − 0.368i)11-s + (0.271 + 0.469i)12-s + (−0.277 + 0.232i)13-s + (−1.27 + 0.462i)14-s + (0.911 + 0.331i)15-s + (0.191 + 0.160i)16-s + (0.100 − 0.570i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.115915 + 0.0791797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.115915 + 0.0791797i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (1.43 + 1.20i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (1.87 - 0.684i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.53 + 4.38i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.705 + 1.22i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 0.839i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.414 + 2.35i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.87 - 1.04i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.46 - 8.32i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.184 + 0.320i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 + (1.17 + 0.984i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.713 + 0.259i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.77 + 10.0i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (1.57 + 0.572i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.124 + 0.705i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (9.17 + 3.33i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.243 + 1.38i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.98 + 2.17i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (3.49 + 2.93i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (1.71 + 1.44i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.99 - 3.45i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.15 - 6.84i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.266 - 1.50i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−9.971521257254313444814897360735, −9.068034243605540167711480838591, −7.63487246505015690922113974739, −7.04392339214717548115903045897, −6.36879797891455912883084793365, −5.02889136905198828021689319363, −3.83930440461322828848833887607, −3.21091454780583331565856517082, −1.13736607828591413039150962736, −0.091271477577683732657987900070,
2.71762246256260058677350193054, 4.08128127429280386628991752965, 4.89152060259858133802161624211, 5.84782714464492560466996378282, 6.33786825353235027058964650927, 7.70132684238619127559231714654, 8.488802936189656381476842970879, 9.412989077563993863609462581520, 10.02503666276557218995106797115, 11.23593572881860882966651709303
