| L(s) = 1 | + 1.73·5-s − 0.517·7-s + 9-s + 1.41·11-s + 1.93·19-s − 1.93·23-s + 1.99·25-s − 29-s − 1.41·31-s − 0.896·35-s − 41-s − 1.93·43-s + 1.73·45-s − 0.732·49-s + 53-s + 2.44·55-s − 1.73·61-s − 0.517·63-s + 0.517·71-s − 0.732·77-s + 81-s + 3.34·95-s − 97-s + 1.41·99-s + 1.41·103-s − 1.73·113-s − 3.34·115-s + ⋯ |
| L(s) = 1 | + 1.73·5-s − 0.517·7-s + 9-s + 1.41·11-s + 1.93·19-s − 1.93·23-s + 1.99·25-s − 29-s − 1.41·31-s − 0.896·35-s − 41-s − 1.93·43-s + 1.73·45-s − 0.732·49-s + 53-s + 2.44·55-s − 1.73·61-s − 0.517·63-s + 0.517·71-s − 0.732·77-s + 81-s + 3.34·95-s − 97-s + 1.41·99-s + 1.41·103-s − 1.73·113-s − 3.34·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.958129906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958129906\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.73T + T^{2} \) |
| 7 | \( 1 + 0.517T + T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.93T + T^{2} \) |
| 23 | \( 1 + 1.93T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + 1.93T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.73T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 0.517T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.053456606400459515221096846404, −7.83782525495293202595087520422, −6.96597079849162135658133720209, −6.47061820407221403157992761793, −5.73477471941363928499750409741, −5.11445292822713014943790207392, −3.95290559826366485004422317270, −3.25851446937475982750635032232, −1.85592598724144471958179504581, −1.49234224594568974162615976694,
1.49234224594568974162615976694, 1.85592598724144471958179504581, 3.25851446937475982750635032232, 3.95290559826366485004422317270, 5.11445292822713014943790207392, 5.73477471941363928499750409741, 6.47061820407221403157992761793, 6.96597079849162135658133720209, 7.83782525495293202595087520422, 9.053456606400459515221096846404
