| L(s) = 1 | − 63.6i·5-s − 524·7-s − 865. i·11-s + 344·13-s + 7.14e3i·17-s + 2.32e3·19-s + 5.75e3i·23-s + 1.15e4·25-s − 2.31e4i·29-s + 1.05e4·31-s + 3.33e4i·35-s − 2.40e4·37-s + 1.08e5i·41-s + 9.09e4·43-s + 1.28e5i·47-s + ⋯ |
| L(s) = 1 | − 0.509i·5-s − 1.52·7-s − 0.650i·11-s + 0.156·13-s + 1.45i·17-s + 0.338·19-s + 0.472i·23-s + 0.740·25-s − 0.949i·29-s + 0.354·31-s + 0.777i·35-s − 0.475·37-s + 1.57i·41-s + 1.14·43-s + 1.24i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.216516605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216516605\) |
| \(L(4)\) |
|
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 | \( 1 + 63.6iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 524T + 1.17e5T^{2} \) |
| 11 | \( 1 + 865. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 344T + 4.82e6T^{2} \) |
| 17 | \( 1 - 7.14e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 2.32e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 5.75e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.31e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.05e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 2.40e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.08e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 9.09e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.28e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.96e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.98e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.51e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.16e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 5.39e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.08e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.40e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.32e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 2.23e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.71e4T + 8.32e11T^{2} \) |
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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
−12.38619384200964713163541735196, −11.06285518294771001089617440636, −9.970624878099726940863214117185, −9.091176547676685476769026020454, −8.024120971737906136729213255414, −6.54712463077993813366066036242, −5.74410181526067886608122137324, −4.07601321271560429900367631506, −2.92515428495083839853314816637, −1.00809641632729382430677895798,
0.45119025551434722694204636363, 2.53893030570814876601858443627, 3.58280034854143074365999037839, 5.20588695502863101123929630959, 6.65366879608939218366303205123, 7.19970216617931207376401411647, 8.908636293553471719664983293792, 9.788215091197648744194926163940, 10.64023916214317475943296051644, 11.94819219920316582682917218961
