| L(s) = 1 | − 7.37i·5-s + 1.26·7-s − 5.82i·11-s − 10.4·13-s − 18.3i·17-s − 20.8·19-s + 20.4i·23-s − 29.4·25-s − 11.1i·29-s + 61.3·31-s − 9.34i·35-s − 38.4·37-s + 33.0i·41-s − 49.3·43-s − 21.5i·47-s + ⋯ |
| L(s) = 1 | − 1.47i·5-s + 0.180·7-s − 0.529i·11-s − 0.806·13-s − 1.07i·17-s − 1.09·19-s + 0.890i·23-s − 1.17·25-s − 0.385i·29-s + 1.97·31-s − 0.266i·35-s − 1.03·37-s + 0.807i·41-s − 1.14·43-s − 0.459i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8077232154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8077232154\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 7.37iT - 25T^{2} \) |
| 7 | \( 1 - 1.26T + 49T^{2} \) |
| 11 | \( 1 + 5.82iT - 121T^{2} \) |
| 13 | \( 1 + 10.4T + 169T^{2} \) |
| 17 | \( 1 + 18.3iT - 289T^{2} \) |
| 19 | \( 1 + 20.8T + 361T^{2} \) |
| 23 | \( 1 - 20.4iT - 529T^{2} \) |
| 29 | \( 1 + 11.1iT - 841T^{2} \) |
| 31 | \( 1 - 61.3T + 961T^{2} \) |
| 37 | \( 1 + 38.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 49.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 21.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 25.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 91.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 114. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 8.21T + 6.24e3T^{2} \) |
| 83 | \( 1 + 150. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 72.8T + 9.40e3T^{2} \) |
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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
−9.504650142515587180090263880538, −8.619708354032375203386299833848, −8.107388488416502439851327509001, −7.02040550135559155310762557725, −5.88619593876182706671158144288, −4.93688857640087056998332056554, −4.40979757213520169153261040079, −2.92662509774570393933209609099, −1.51324066690022645921240430737, −0.25659421368053854390674822334,
1.96046035274488664661615819744, 2.86565008736382194231257202476, 4.00933761571991179900017938507, 5.03948347179334508897889582175, 6.49828626600065738269363285512, 6.67600074884769577538899225736, 7.82427825265752861306532623169, 8.564200276568251982432385452103, 9.862915810209248310694454483426, 10.40618859224512183716562991614
