| L(s) = 1 | − 3·5-s − 5·7-s − 9·11-s + 6·13-s − 12·17-s − 9·23-s + 5·25-s + 18·29-s + 9·31-s + 15·35-s + 2·37-s + 3·41-s − 10·43-s + 6·47-s + 18·49-s + 27·55-s − 6·59-s + 24·61-s − 18·65-s − 2·67-s + 45·77-s − 14·79-s − 6·83-s + 36·85-s + 18·89-s − 30·91-s + 12·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.88·7-s − 2.71·11-s + 1.66·13-s − 2.91·17-s − 1.87·23-s + 25-s + 3.34·29-s + 1.61·31-s + 2.53·35-s + 0.328·37-s + 0.468·41-s − 1.52·43-s + 0.875·47-s + 18/7·49-s + 3.64·55-s − 0.781·59-s + 3.07·61-s − 2.23·65-s − 0.244·67-s + 5.12·77-s − 1.57·79-s − 0.658·83-s + 3.90·85-s + 1.90·89-s − 3.14·91-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2383327905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2383327905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 18 T + 137 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.965582134882019551639942972133, −8.754164703075188022570699082981, −8.257543416871809701758181513378, −8.088736866739174323397495007089, −8.069146415829963395603865853482, −7.11106203635024905451752668965, −6.73047747157985758922350490261, −6.72620097489289444206399964457, −6.00745279469892424783902403835, −5.99582985030100934818224498391, −5.20656565635366265092621270037, −4.61882487654570121696261685868, −4.36452971064340424430364140235, −4.00609325707489503579030891712, −3.42486199155014047932433868534, −2.92345258787829591453167889765, −2.52621252122188378306326033244, −2.31849614996403011751768053581, −0.894959951490806760427160439264, −0.21245678210612668012323254067,
0.21245678210612668012323254067, 0.894959951490806760427160439264, 2.31849614996403011751768053581, 2.52621252122188378306326033244, 2.92345258787829591453167889765, 3.42486199155014047932433868534, 4.00609325707489503579030891712, 4.36452971064340424430364140235, 4.61882487654570121696261685868, 5.20656565635366265092621270037, 5.99582985030100934818224498391, 6.00745279469892424783902403835, 6.72620097489289444206399964457, 6.73047747157985758922350490261, 7.11106203635024905451752668965, 8.069146415829963395603865853482, 8.088736866739174323397495007089, 8.257543416871809701758181513378, 8.754164703075188022570699082981, 8.965582134882019551639942972133
