| L(s) = 1 | + 1.73i·3-s + 4.92·5-s + 2.92i·7-s − 2.99·9-s + 14.9i·11-s + 23.8·13-s + 8.53i·15-s − 19.8·17-s − 30.9i·19-s − 5.07·21-s − 8i·23-s − 0.712·25-s − 5.19i·27-s − 16.9·29-s − 38.6i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.985·5-s + 0.418i·7-s − 0.333·9-s + 1.35i·11-s + 1.83·13-s + 0.569i·15-s − 1.16·17-s − 1.62i·19-s − 0.241·21-s − 0.347i·23-s − 0.0285·25-s − 0.192i·27-s − 0.583·29-s − 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.39059 + 0.576001i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39059 + 0.576001i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| good | 5 | \( 1 - 4.92T + 25T^{2} \) |
| 7 | \( 1 - 2.92iT - 49T^{2} \) |
| 11 | \( 1 - 14.9iT - 121T^{2} \) |
| 13 | \( 1 - 23.8T + 169T^{2} \) |
| 17 | \( 1 + 19.8T + 289T^{2} \) |
| 19 | \( 1 + 30.9iT - 361T^{2} \) |
| 23 | \( 1 + 8iT - 529T^{2} \) |
| 29 | \( 1 + 16.9T + 841T^{2} \) |
| 31 | \( 1 + 38.6iT - 961T^{2} \) |
| 37 | \( 1 + 9.71T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.14T + 1.68e3T^{2} \) |
| 43 | \( 1 + 5.35iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 69.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.928T + 2.80e3T^{2} \) |
| 59 | \( 1 - 108. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 14T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 43.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 25.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 62.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 50.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 145.T + 9.40e3T^{2} \) |
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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
−13.65357163078885824448068401662, −13.10332901054050590454051047001, −11.55317734569708828927557017344, −10.57685640021272265820808884074, −9.435506095578784216893133493786, −8.711146598929006720556000969288, −6.82643776619145494083961770822, −5.65345616140089901493346566272, −4.26862515575823178487598376000, −2.23618549646777823342743772722,
1.50232231956159241405372414717, 3.56343940088890762983704225215, 5.76144643923879782471329869141, 6.43560936907700304818563116303, 8.119192369434681752774494082333, 9.049655692583855219860091723286, 10.54217149815033758427005081219, 11.31632877312068901413057685502, 12.83157071480969689466237406970, 13.72054997847265083519025111610
