L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 4·13-s + 5·16-s + 4·19-s − 6·23-s − 10·25-s − 8·26-s + 10·31-s − 6·32-s − 8·37-s − 8·38-s − 6·41-s + 4·43-s + 12·46-s − 18·47-s + 20·50-s + 12·52-s + 4·61-s − 20·62-s + 7·64-s − 8·67-s + 6·71-s − 14·73-s + 16·74-s + 12·76-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1.10·13-s + 5/4·16-s + 0.917·19-s − 1.25·23-s − 2·25-s − 1.56·26-s + 1.79·31-s − 1.06·32-s − 1.31·37-s − 1.29·38-s − 0.937·41-s + 0.609·43-s + 1.76·46-s − 2.62·47-s + 2.82·50-s + 1.66·52-s + 0.512·61-s − 2.54·62-s + 7/8·64-s − 0.977·67-s + 0.712·71-s − 1.63·73-s + 1.85·74-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 157 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 292 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 115 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.78312503786054005380649725538, −7.44547157283786570339904366446, −6.92763269346115620758097124197, −6.89072610232082256318237503192, −6.26931087984648627936420138201, −6.08383797806554069085669540053, −5.63794531073983400258737179861, −5.58225875109505807275131568194, −4.74773051448940837678578365718, −4.57683945792974113373398514337, −3.89933037120271914288259442941, −3.67308884737906399938068042142, −3.10267295388521042223410723807, −2.98821678416405042379509172268, −2.05866046415752733404487526986, −2.04407615467767609079328752608, −1.24461894250784106165013342370, −1.21110064269771303972230508226, 0, 0,
1.21110064269771303972230508226, 1.24461894250784106165013342370, 2.04407615467767609079328752608, 2.05866046415752733404487526986, 2.98821678416405042379509172268, 3.10267295388521042223410723807, 3.67308884737906399938068042142, 3.89933037120271914288259442941, 4.57683945792974113373398514337, 4.74773051448940837678578365718, 5.58225875109505807275131568194, 5.63794531073983400258737179861, 6.08383797806554069085669540053, 6.26931087984648627936420138201, 6.89072610232082256318237503192, 6.92763269346115620758097124197, 7.44547157283786570339904366446, 7.78312503786054005380649725538