L(s) = 1 | − 8.11i·5-s + 9.48i·7-s + 10.5·11-s + 20.9i·13-s − 4.92·17-s − 26.9·19-s − 43.7i·23-s − 40.9·25-s + 14.5i·29-s − 25.4i·31-s + 77.0·35-s + 12.9i·37-s − 50.1·41-s − 5.03·43-s − 40.8i·47-s + ⋯ |
L(s) = 1 | − 1.62i·5-s + 1.35i·7-s + 0.962·11-s + 1.61i·13-s − 0.289·17-s − 1.41·19-s − 1.90i·23-s − 1.63·25-s + 0.500i·29-s − 0.822i·31-s + 2.20·35-s + 0.350i·37-s − 1.22·41-s − 0.117·43-s − 0.869i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9688384016\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9688384016\) |
\(L(2)\) |
|
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 | \( 1 + 8.11iT - 25T^{2} \) |
| 7 | \( 1 - 9.48iT - 49T^{2} \) |
| 11 | \( 1 - 10.5T + 121T^{2} \) |
| 13 | \( 1 - 20.9iT - 169T^{2} \) |
| 17 | \( 1 + 4.92T + 289T^{2} \) |
| 19 | \( 1 + 26.9T + 361T^{2} \) |
| 23 | \( 1 + 43.7iT - 529T^{2} \) |
| 29 | \( 1 - 14.5iT - 841T^{2} \) |
| 31 | \( 1 + 25.4iT - 961T^{2} \) |
| 37 | \( 1 - 12.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 50.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 5.03T + 1.84e3T^{2} \) |
| 47 | \( 1 + 40.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 27.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 28.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 65.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 127.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 35.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 101.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 15.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 73.8T + 9.40e3T^{2} \) |
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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
−8.641144126242277756292247090546, −8.250246834241070899075128584409, −6.61377943417023197965271904881, −6.36820905995697151200839913345, −5.19299648541856360303852584229, −4.58192411599056834969215177394, −3.91681165727402093776885562270, −2.25598138494463696018257223005, −1.68700479650629806455347407208, −0.23828766475439324093409652771,
1.19077855987142597990982206702, 2.50938415484935028289251819199, 3.57902667409893119649344269836, 3.86077169879976808699993184142, 5.21880130184474049827022984630, 6.31864125273760542583229907516, 6.74706438868782969206275220703, 7.53281032258576913223759662556, 8.044609505423430452800072044718, 9.252592103890068795260897586604