L(s) = 1 | + (0.195 − 0.980i)2-s + (−0.923 − 0.382i)4-s + (0.980 − 0.195i)5-s + (−0.555 + 0.831i)8-s − i·10-s + (0.707 + 0.707i)16-s + (0.275 − 0.275i)17-s + (0.216 − 1.08i)19-s + (−0.980 − 0.195i)20-s + (−0.360 − 0.149i)23-s + (0.923 − 0.382i)25-s + 0.765·31-s + (0.831 − 0.555i)32-s + (−0.216 − 0.324i)34-s + (−1.02 − 0.425i)38-s + ⋯ |
L(s) = 1 | + (0.195 − 0.980i)2-s + (−0.923 − 0.382i)4-s + (0.980 − 0.195i)5-s + (−0.555 + 0.831i)8-s − i·10-s + (0.707 + 0.707i)16-s + (0.275 − 0.275i)17-s + (0.216 − 1.08i)19-s + (−0.980 − 0.195i)20-s + (−0.360 − 0.149i)23-s + (0.923 − 0.382i)25-s + 0.765·31-s + (0.831 − 0.555i)32-s + (−0.216 − 0.324i)34-s + (−1.02 − 0.425i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.449632723\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.449632723\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.195 + 0.980i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.980 + 0.195i)T \) |
good | 7 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 17 | \( 1 + (-0.275 + 0.275i)T - iT^{2} \) |
| 19 | \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \) |
| 23 | \( 1 + (0.360 + 0.149i)T + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 - 0.765T + T^{2} \) |
| 37 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 47 | \( 1 + (1.17 + 1.17i)T + iT^{2} \) |
| 53 | \( 1 + (-0.425 + 0.636i)T + (-0.382 - 0.923i)T^{2} \) |
| 59 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 61 | \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (-1 - i)T + iT^{2} \) |
| 83 | \( 1 + (1.81 + 0.360i)T + (0.923 + 0.382i)T^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.833566475842304817866897666987, −8.386720729618673554761170061905, −7.13542342898798721357409164704, −6.28352054518500943376987264412, −5.38967481876352892454143250051, −4.88863668361951704718900342249, −3.88752705388226952062975612285, −2.82817851513035016507267637672, −2.13472417180944745015812404986, −0.970293690920145369487586899141,
1.43410570224956498401158847574, 2.79344821455558948114929875385, 3.75778753487510310125775440283, 4.71427766147394240544939091377, 5.57344680239494113374215131791, 6.09878836324473637457865486522, 6.75614343056475302995062757470, 7.67519834681296054237947888171, 8.269061982162761179117217909543, 9.120925054211771796423963102063