| L(s) = 1 | + i·3-s + 3.84·5-s + (−1.62 + 2.09i)7-s − 9-s + 4.54·11-s − 1.81·13-s + 3.84i·15-s + 3.49i·17-s + 1.68i·19-s + (−2.09 − 1.62i)21-s − 5.00i·23-s + 9.77·25-s − i·27-s − 1.81i·29-s − 5.34·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.71·5-s + (−0.612 + 0.790i)7-s − 0.333·9-s + 1.37·11-s − 0.503·13-s + 0.992i·15-s + 0.846i·17-s + 0.387i·19-s + (−0.456 − 0.353i)21-s − 1.04i·23-s + 1.95·25-s − 0.192i·27-s − 0.337i·29-s − 0.959·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67087 + 0.959709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67087 + 0.959709i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (1.62 - 2.09i)T \) |
| good | 5 | \( 1 - 3.84T + 5T^{2} \) |
| 11 | \( 1 - 4.54T + 11T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 - 3.49iT - 17T^{2} \) |
| 19 | \( 1 - 1.68iT - 19T^{2} \) |
| 23 | \( 1 + 5.00iT - 23T^{2} \) |
| 29 | \( 1 + 1.81iT - 29T^{2} \) |
| 31 | \( 1 + 5.34T + 31T^{2} \) |
| 37 | \( 1 + 1.42iT - 37T^{2} \) |
| 41 | \( 1 - 8.97iT - 41T^{2} \) |
| 43 | \( 1 - 8.03T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 - 5.87iT - 53T^{2} \) |
| 59 | \( 1 + 8.46iT - 59T^{2} \) |
| 61 | \( 1 - 3.01T + 61T^{2} \) |
| 67 | \( 1 - 4.42T + 67T^{2} \) |
| 71 | \( 1 + 1.47iT - 71T^{2} \) |
| 73 | \( 1 - 6.98iT - 73T^{2} \) |
| 79 | \( 1 - 2.97iT - 79T^{2} \) |
| 83 | \( 1 - 10.5iT - 83T^{2} \) |
| 89 | \( 1 + 15.9iT - 89T^{2} \) |
| 97 | \( 1 + 11.6iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.37913887789423204448352859695, −9.626065980164858505528162337058, −9.269378317516344098972809311382, −8.401750933327947714322815925089, −6.70147697400932073821576196454, −6.11946190246203053697551707234, −5.41332551718978635243509417504, −4.16695637021058184253229523128, −2.83259679820269402388588033862, −1.75331411038597706957004715394,
1.15481979653014895334284370034, 2.30890378953675218799944678428, 3.61282267600652963232742881542, 5.09054959579593262040955784330, 6.02268352763493652800330405665, 6.80930717853253048744352746310, 7.38298649777219496462610407883, 9.058218239383114410245745981674, 9.384555878300692522830326948475, 10.19621879247718152290609279649
