| L(s) = 1 | + 4·5-s + 4·7-s − 2·13-s + 10·17-s + 8·19-s − 4·23-s + 11·25-s + 16·35-s + 2·37-s − 12·43-s + 4·47-s + 8·49-s + 14·53-s − 8·59-s − 8·61-s − 8·65-s − 20·67-s − 6·73-s + 32·79-s + 4·83-s + 40·85-s − 8·91-s + 32·95-s − 6·97-s + 12·101-s − 12·103-s − 12·107-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s − 0.554·13-s + 2.42·17-s + 1.83·19-s − 0.834·23-s + 11/5·25-s + 2.70·35-s + 0.328·37-s − 1.82·43-s + 0.583·47-s + 8/7·49-s + 1.92·53-s − 1.04·59-s − 1.02·61-s − 0.992·65-s − 2.44·67-s − 0.702·73-s + 3.60·79-s + 0.439·83-s + 4.33·85-s − 0.838·91-s + 3.28·95-s − 0.609·97-s + 1.19·101-s − 1.18·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.099811914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.099811914\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.639725237208135824646120450139, −9.568859221961153784326558872765, −8.956764929303487065657844439251, −8.614283874811762529455210488229, −7.906595688678972229622209256425, −7.86779084098243298372782831738, −7.31470080386650773782382785280, −7.09651170592945637339036943800, −6.15586633385312195368740810358, −6.04484554736855802635706115914, −5.44184412217736538359608171911, −5.31720677068749539916046171645, −4.86211797511930658999253024370, −4.47385662067453532093154674080, −3.44854237484596419268831412485, −3.28296345776681792012266116063, −2.49485738791724596378821822991, −2.00354792028057149372389013836, −1.32883875504121738721873184738, −1.10234636347963754766446455369,
1.10234636347963754766446455369, 1.32883875504121738721873184738, 2.00354792028057149372389013836, 2.49485738791724596378821822991, 3.28296345776681792012266116063, 3.44854237484596419268831412485, 4.47385662067453532093154674080, 4.86211797511930658999253024370, 5.31720677068749539916046171645, 5.44184412217736538359608171911, 6.04484554736855802635706115914, 6.15586633385312195368740810358, 7.09651170592945637339036943800, 7.31470080386650773782382785280, 7.86779084098243298372782831738, 7.906595688678972229622209256425, 8.614283874811762529455210488229, 8.956764929303487065657844439251, 9.568859221961153784326558872765, 9.639725237208135824646120450139
