| L(s) = 1 | + 5-s + 1.73·7-s + 9-s − 1.73·11-s − 13-s − 1.73·23-s + 1.73·35-s − 41-s + 45-s + 1.99·49-s − 1.73·55-s + 1.73·59-s − 61-s + 1.73·63-s − 65-s + 1.73·67-s + 73-s − 2.99·77-s − 1.73·79-s + 81-s − 1.73·91-s − 2·97-s − 1.73·99-s − 109-s − 113-s − 1.73·115-s − 117-s + ⋯ |
| L(s) = 1 | + 5-s + 1.73·7-s + 9-s − 1.73·11-s − 13-s − 1.73·23-s + 1.73·35-s − 41-s + 45-s + 1.99·49-s − 1.73·55-s + 1.73·59-s − 61-s + 1.73·63-s − 65-s + 1.73·67-s + 73-s − 2.99·77-s − 1.73·79-s + 81-s − 1.73·91-s − 2·97-s − 1.73·99-s − 109-s − 113-s − 1.73·115-s − 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.300882212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300882212\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.73T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.73T + T^{2} \) |
| 67 | \( 1 - 1.73T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−10.14853736181523255969087843639, −9.681854634421752525365627571264, −8.267353686550866216673289175350, −7.85957501973448490714394140539, −6.96895555370816725001799748903, −5.59376612261442446367254223207, −5.10748209512788934814198373082, −4.23212097657041214622301498662, −2.39582097143609153951770010223, −1.77500013501147469243296806692,
1.77500013501147469243296806692, 2.39582097143609153951770010223, 4.23212097657041214622301498662, 5.10748209512788934814198373082, 5.59376612261442446367254223207, 6.96895555370816725001799748903, 7.85957501973448490714394140539, 8.267353686550866216673289175350, 9.681854634421752525365627571264, 10.14853736181523255969087843639
