L(s) = 1 | + (−1.37 − 0.320i)2-s + (−0.720 − 0.720i)3-s + (1.79 + 0.883i)4-s + (0.707 − 0.707i)5-s + (0.761 + 1.22i)6-s − 4.02i·7-s + (−2.18 − 1.79i)8-s − 1.96i·9-s + (−1.20 + 0.747i)10-s + (−0.646 + 0.646i)11-s + (−0.656 − 1.92i)12-s + (4.91 + 4.91i)13-s + (−1.29 + 5.54i)14-s − 1.01·15-s + (2.43 + 3.17i)16-s − 2.70·17-s + ⋯ |
L(s) = 1 | + (−0.973 − 0.226i)2-s + (−0.416 − 0.416i)3-s + (0.897 + 0.441i)4-s + (0.316 − 0.316i)5-s + (0.310 + 0.499i)6-s − 1.52i·7-s + (−0.773 − 0.633i)8-s − 0.653i·9-s + (−0.379 + 0.236i)10-s + (−0.195 + 0.195i)11-s + (−0.189 − 0.557i)12-s + (1.36 + 1.36i)13-s + (−0.345 + 1.48i)14-s − 0.263·15-s + (0.609 + 0.792i)16-s − 0.656·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.474312 - 0.363579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.474312 - 0.363579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 + 0.320i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (0.720 + 0.720i)T + 3iT^{2} \) |
| 7 | \( 1 + 4.02iT - 7T^{2} \) |
| 11 | \( 1 + (0.646 - 0.646i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.91 - 4.91i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + (0.438 + 0.438i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.60iT - 23T^{2} \) |
| 29 | \( 1 + (-2.00 - 2.00i)T + 29iT^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 + (0.743 - 0.743i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.603iT - 41T^{2} \) |
| 43 | \( 1 + (5.03 - 5.03i)T - 43iT^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + (-4.07 + 4.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.22 + 1.22i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.98 + 6.98i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.24 - 5.24i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.7iT - 71T^{2} \) |
| 73 | \( 1 + 1.30iT - 73T^{2} \) |
| 79 | \( 1 - 0.611T + 79T^{2} \) |
| 83 | \( 1 + (-1.29 - 1.29i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.9iT - 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−13.92241921590225480842768759799, −13.01570396244850904397974789164, −11.69984601475331558247888847965, −10.89132235413497354020025377869, −9.721073725190053844788809779511, −8.619824817587449666388945297995, −7.12702016261729048507604825085, −6.37985419476813256100835789574, −3.94498258765149888742711374704, −1.27794611150868417872909509578,
2.56040380100212905166033433798, 5.45035496836055385583681898823, 6.22957795488557128833993290771, 8.101286405768503335422096350196, 8.903093601010574580808893302832, 10.33107372396402994937059379504, 10.95289056152024646132392210821, 12.14305616995706872113181182057, 13.60374074808582237255155971363, 15.31086868097340381268975576101