| L(s) = 1 | − 2.04·3-s − 8.44i·5-s − 6.75i·7-s − 4.83·9-s − 9.79·11-s − 20.6i·13-s + 17.2i·15-s − 29.6·17-s − 4.35·19-s + 13.7i·21-s + 3.39i·23-s − 46.3·25-s + 28.2·27-s + 6.59i·29-s − 25.7i·31-s + ⋯ |
| L(s) = 1 | − 0.680·3-s − 1.68i·5-s − 0.965i·7-s − 0.537·9-s − 0.890·11-s − 1.59i·13-s + 1.14i·15-s − 1.74·17-s − 0.229·19-s + 0.656i·21-s + 0.147i·23-s − 1.85·25-s + 1.04·27-s + 0.227i·29-s − 0.830i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5167512083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5167512083\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35T \) |
| good | 3 | \( 1 + 2.04T + 9T^{2} \) |
| 5 | \( 1 + 8.44iT - 25T^{2} \) |
| 7 | \( 1 + 6.75iT - 49T^{2} \) |
| 11 | \( 1 + 9.79T + 121T^{2} \) |
| 13 | \( 1 + 20.6iT - 169T^{2} \) |
| 17 | \( 1 + 29.6T + 289T^{2} \) |
| 23 | \( 1 - 3.39iT - 529T^{2} \) |
| 29 | \( 1 - 6.59iT - 841T^{2} \) |
| 31 | \( 1 + 25.7iT - 961T^{2} \) |
| 37 | \( 1 + 20.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 74.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 13.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 13.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 24.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 74.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 12.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 46.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 73.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 80.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 29.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 116.T + 9.40e3T^{2} \) |
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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
−8.875184470689352719080190493548, −8.174082968531868103813966534391, −7.50188197887955324991621554835, −6.20810029608225179769413857789, −5.41690528232072014127488052463, −4.82465074281592761895364799995, −3.95083492072477436673698822833, −2.42873782802354415744366800097, −0.78030937887276723344239177082, −0.22902035287198234270542725355,
2.28605056384589536342048984230, 2.66638815891675252448195465268, 4.09128655452023910583467578346, 5.18185843758647952807056274740, 6.22239785983357852652938611488, 6.55701926976101405451017406582, 7.42282300906331332832452536020, 8.627569220637660794996150068640, 9.245104848296820490793575803940, 10.44600435503468031375530666986
