| L(s) = 1 | − 5·5-s + 2·7-s + 5·11-s + 6·13-s − 4·17-s − 2·19-s − 2·23-s + 10·25-s − 5·29-s − 10·35-s + 11·37-s + 41-s + 7·43-s + 17·47-s + 3·49-s + 9·53-s − 25·55-s + 3·59-s − 9·61-s − 30·65-s − 12·67-s − 7·71-s + 22·73-s + 10·77-s − 7·79-s − 8·83-s + 20·85-s + ⋯ |
| L(s) = 1 | − 2.23·5-s + 0.755·7-s + 1.50·11-s + 1.66·13-s − 0.970·17-s − 0.458·19-s − 0.417·23-s + 2·25-s − 0.928·29-s − 1.69·35-s + 1.80·37-s + 0.156·41-s + 1.06·43-s + 2.47·47-s + 3/7·49-s + 1.23·53-s − 3.37·55-s + 0.390·59-s − 1.15·61-s − 3.72·65-s − 1.46·67-s − 0.830·71-s + 2.57·73-s + 1.13·77-s − 0.787·79-s − 0.878·83-s + 2.16·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.473613653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.473613653\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 73 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 51 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 17 T + 155 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 125 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 89 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 131 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 53 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 262 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 159 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 167 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 13 T + 135 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−7.78142585632911147793892515263, −7.56227729721006237410556702592, −7.21770093575038299974209531980, −7.09896328351569422154828926129, −6.33167366566478104734830640402, −6.30473310746095811008393702191, −5.80406933911011178746656381321, −5.69518747334058072905459953542, −4.84646330250820283846276894745, −4.56491387738026598295641692599, −4.18608526971003170217114372318, −4.03540046908232319304220335937, −3.74752380697299152795644095129, −3.59643036998779116244082795060, −2.87552512628305221414259723137, −2.40137028356060628475981772526, −1.91986732514385655630261344307, −1.34482670994166467909836355691, −0.848867351532436814216510324145, −0.47893153494190931553087719898,
0.47893153494190931553087719898, 0.848867351532436814216510324145, 1.34482670994166467909836355691, 1.91986732514385655630261344307, 2.40137028356060628475981772526, 2.87552512628305221414259723137, 3.59643036998779116244082795060, 3.74752380697299152795644095129, 4.03540046908232319304220335937, 4.18608526971003170217114372318, 4.56491387738026598295641692599, 4.84646330250820283846276894745, 5.69518747334058072905459953542, 5.80406933911011178746656381321, 6.30473310746095811008393702191, 6.33167366566478104734830640402, 7.09896328351569422154828926129, 7.21770093575038299974209531980, 7.56227729721006237410556702592, 7.78142585632911147793892515263
