| L(s) = 1 | + 2-s + 4·4-s + 5·5-s + 11·8-s + 5·10-s + 11·16-s + 28·17-s − 44·19-s + 20·20-s + 34·23-s − 2·31-s + 44·32-s + 28·34-s − 44·38-s + 55·40-s + 34·46-s − 14·47-s − 49·49-s + 172·53-s + 118·61-s − 2·62-s + 57·64-s + 112·68-s − 176·76-s − 98·79-s + 55·80-s + 154·83-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 4-s + 5-s + 11/8·8-s + 1/2·10-s + 0.687·16-s + 1.64·17-s − 2.31·19-s + 20-s + 1.47·23-s − 0.0645·31-s + 11/8·32-s + 0.823·34-s − 1.15·38-s + 11/8·40-s + 0.739·46-s − 0.297·47-s − 49-s + 3.24·53-s + 1.93·61-s − 0.0322·62-s + 0.890·64-s + 1.64·68-s − 2.31·76-s − 1.24·79-s + 0.687·80-s + 1.85·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.538668082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.538668082\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - 3 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T + 627 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 957 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T - 2013 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 86 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 118 T + 10203 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 98 T + 3363 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T + 16827 T^{2} - 154 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
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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
−11.17708905180259586583681831667, −10.71561776157807423894264130213, −10.47538770597676681047998078786, −10.10479142141370149833633487312, −9.566600354970446415429227173205, −9.056464400333833658075718421714, −8.305045179061261151664594749917, −8.161177935817064242033359215539, −7.34223680721777854611560757242, −6.94471855151121318315444837593, −6.56808034587258515696422234014, −6.05520022633615213076379536416, −5.36471371817810702952132776628, −5.21181337918442184444969493154, −4.29164944346108434254782178622, −3.88401710049670527581232413743, −2.97176544067916999604825432161, −2.34829681168297189790167771744, −1.83216309759707665518493366196, −0.994379589510945217032105007996,
0.994379589510945217032105007996, 1.83216309759707665518493366196, 2.34829681168297189790167771744, 2.97176544067916999604825432161, 3.88401710049670527581232413743, 4.29164944346108434254782178622, 5.21181337918442184444969493154, 5.36471371817810702952132776628, 6.05520022633615213076379536416, 6.56808034587258515696422234014, 6.94471855151121318315444837593, 7.34223680721777854611560757242, 8.161177935817064242033359215539, 8.305045179061261151664594749917, 9.056464400333833658075718421714, 9.566600354970446415429227173205, 10.10479142141370149833633487312, 10.47538770597676681047998078786, 10.71561776157807423894264130213, 11.17708905180259586583681831667
