| L(s) = 1 | − 2·5-s − 10·7-s + 10·9-s − 10·11-s − 50·17-s − 38·19-s + 20·23-s − 47·25-s + 20·35-s − 10·43-s − 20·45-s − 10·47-s − 23·49-s + 20·55-s + 190·61-s − 100·63-s − 50·73-s + 100·77-s + 19·81-s + 260·83-s + 100·85-s + 76·95-s − 100·99-s + 100·101-s − 40·115-s + 500·119-s − 167·121-s + ⋯ |
| L(s) = 1 | − 2/5·5-s − 1.42·7-s + 10/9·9-s − 0.909·11-s − 2.94·17-s − 2·19-s + 0.869·23-s − 1.87·25-s + 4/7·35-s − 0.232·43-s − 4/9·45-s − 0.212·47-s − 0.469·49-s + 4/11·55-s + 3.11·61-s − 1.58·63-s − 0.684·73-s + 1.29·77-s + 0.234·81-s + 3.13·83-s + 1.17·85-s + 4/5·95-s − 1.01·99-s + 0.990·101-s − 0.347·115-s + 4.20·119-s − 1.38·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3641440986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3641440986\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 10 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 118 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 122 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2090 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 1562 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 4970 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 95 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 3190 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10682 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 358 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18530 T^{2} + p^{4} T^{4} \) |
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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
−11.70832563362660317317337754457, −11.02125266818162783581775244859, −10.95794059830453745703831634225, −10.27195752093586346980793830614, −9.929832168873740650870943323342, −9.399008423768557816554457060099, −8.977335498184274514108441299337, −8.358246437050222303638081471313, −8.050747824597529460200638406356, −7.05652417254592547482618923282, −7.00215994013414804572339390639, −6.34664494505104264980815548554, −6.10112240731783460707151321019, −5.02011616635364706586195241331, −4.52428916598865665784564564396, −4.00350686533231408432810097176, −3.48007248275572484484663990982, −2.34385693728191459883639534368, −2.09586188986499122709036501009, −0.27301913763189484615721775473,
0.27301913763189484615721775473, 2.09586188986499122709036501009, 2.34385693728191459883639534368, 3.48007248275572484484663990982, 4.00350686533231408432810097176, 4.52428916598865665784564564396, 5.02011616635364706586195241331, 6.10112240731783460707151321019, 6.34664494505104264980815548554, 7.00215994013414804572339390639, 7.05652417254592547482618923282, 8.050747824597529460200638406356, 8.358246437050222303638081471313, 8.977335498184274514108441299337, 9.399008423768557816554457060099, 9.929832168873740650870943323342, 10.27195752093586346980793830614, 10.95794059830453745703831634225, 11.02125266818162783581775244859, 11.70832563362660317317337754457
