| L(s) = 1 | + 2·5-s + 6·7-s − 2·13-s + 2·19-s − 4·23-s − 2·25-s + 8·29-s + 14·31-s + 12·35-s + 14·41-s + 4·43-s − 8·47-s + 18·49-s + 8·53-s − 4·59-s + 16·61-s − 4·65-s − 2·67-s − 20·71-s − 12·83-s + 18·89-s − 12·91-s + 4·95-s + 24·97-s − 20·101-s + 4·103-s − 4·107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.26·7-s − 0.554·13-s + 0.458·19-s − 0.834·23-s − 2/5·25-s + 1.48·29-s + 2.51·31-s + 2.02·35-s + 2.18·41-s + 0.609·43-s − 1.16·47-s + 18/7·49-s + 1.09·53-s − 0.520·59-s + 2.04·61-s − 0.496·65-s − 0.244·67-s − 2.37·71-s − 1.31·83-s + 1.90·89-s − 1.25·91-s + 0.410·95-s + 2.43·97-s − 1.99·101-s + 0.394·103-s − 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.730593543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.730593543\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 14 T + 106 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.558778459776546118642854288772, −8.411706726144542245121671181473, −7.81905200637318942706002531313, −7.79625372389190596718912655000, −7.36291747471974795471953154278, −6.91180031744610243025595520400, −6.27150529489126816584147048443, −6.14185501284537459441623711302, −5.70369379927494741227301311202, −5.28230081568859977428875953484, −4.82003791747613209871908947180, −4.65663808892021734152891098265, −4.23810433234328933558510076005, −3.89097246789153798383020306462, −2.88006693389607723093110688272, −2.75252707078851905968961250189, −2.10728983070881774434810393433, −1.85545507379180417618131183004, −1.13432136684212882807748148975, −0.800843534661357137862470340910,
0.800843534661357137862470340910, 1.13432136684212882807748148975, 1.85545507379180417618131183004, 2.10728983070881774434810393433, 2.75252707078851905968961250189, 2.88006693389607723093110688272, 3.89097246789153798383020306462, 4.23810433234328933558510076005, 4.65663808892021734152891098265, 4.82003791747613209871908947180, 5.28230081568859977428875953484, 5.70369379927494741227301311202, 6.14185501284537459441623711302, 6.27150529489126816584147048443, 6.91180031744610243025595520400, 7.36291747471974795471953154278, 7.79625372389190596718912655000, 7.81905200637318942706002531313, 8.411706726144542245121671181473, 8.558778459776546118642854288772
