L(s) = 1 | + 3.74·3-s − 5·5-s + 9.79i·7-s + 5·9-s + 4·11-s + 11.2·13-s − 18.7·15-s + 19.5i·17-s + (−5 + 18.3i)19-s + 36.6i·21-s − 9.79i·23-s + 25·25-s − 14.9·27-s + 36.6i·29-s + 36.6i·31-s + ⋯ |
L(s) = 1 | + 1.24·3-s − 5-s + 1.39i·7-s + 0.555·9-s + 0.363·11-s + 0.863·13-s − 1.24·15-s + 1.15i·17-s + (−0.263 + 0.964i)19-s + 1.74i·21-s − 0.425i·23-s + 25-s − 0.554·27-s + 1.26i·29-s + 1.18i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.60439 + 1.22537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60439 + 1.22537i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 19 | \( 1 + (5 - 18.3i)T \) |
good | 3 | \( 1 - 3.74T + 9T^{2} \) |
| 7 | \( 1 - 9.79iT - 49T^{2} \) |
| 11 | \( 1 - 4T + 121T^{2} \) |
| 13 | \( 1 - 11.2T + 169T^{2} \) |
| 17 | \( 1 - 19.5iT - 289T^{2} \) |
| 23 | \( 1 + 9.79iT - 529T^{2} \) |
| 29 | \( 1 - 36.6iT - 841T^{2} \) |
| 31 | \( 1 - 36.6iT - 961T^{2} \) |
| 37 | \( 1 - 33.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 36.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 9.79iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 56.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 100T + 3.72e3T^{2} \) |
| 67 | \( 1 - 11.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 36.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 19.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 29.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 123.T + 9.40e3T^{2} \) |
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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
−11.40675332052595630210289551379, −10.39707271077718408252590701514, −9.001990206018308826497843219134, −8.590440903846923453344943411773, −8.028384187159167404224475557036, −6.69300177185083550730662522146, −5.51332006375259276678092410948, −3.94371164061317452962283155415, −3.21085231723320378163872647155, −1.86660518366274721539016618816,
0.811278483283539593880593572809, 2.79074090318952672253101797640, 3.82225869341339774614690558526, 4.50935786954804357446592381000, 6.47230037347104298823465827339, 7.57158604575284248828644447285, 7.953810778608334633139076087399, 9.062994830567694338987297831991, 9.823886014639949321630000940608, 11.17841013757230606725521920851