| L(s) = 1 | − 3·3-s + 5-s + 3·9-s + 5·11-s + 10·13-s − 3·15-s − 7·17-s − 2·19-s + 2·23-s + 14·29-s + 4·31-s − 15·33-s + 6·37-s − 30·39-s + 24·41-s − 4·43-s + 3·45-s + 47-s + 21·51-s + 5·55-s + 6·57-s − 4·59-s + 4·61-s + 10·65-s − 8·67-s − 6·69-s + 6·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 9-s + 1.50·11-s + 2.77·13-s − 0.774·15-s − 1.69·17-s − 0.458·19-s + 0.417·23-s + 2.59·29-s + 0.718·31-s − 2.61·33-s + 0.986·37-s − 4.80·39-s + 3.74·41-s − 0.609·43-s + 0.447·45-s + 0.145·47-s + 2.94·51-s + 0.674·55-s + 0.794·57-s − 0.520·59-s + 0.512·61-s + 1.24·65-s − 0.977·67-s − 0.722·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.956950036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.956950036\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.419405070665591639616921151943, −8.859469700794821283040768940734, −8.820918428669247315650459461074, −8.179772671739186686735537662119, −8.023789359013322435640585830623, −7.15523333425596544443829056455, −6.60295424472382285492880394567, −6.39723349026311220489765572830, −6.29364345372281787405357620537, −6.03776595078994905001827323637, −5.51779709043169992480590680719, −5.02125871082504210783273321456, −4.29098332677716516133742311035, −4.24201291488653967627097366766, −3.89863924625019302538537302894, −2.92305349634088367619143291018, −2.58696421747571824449021480651, −1.62458168181415894340280030754, −1.04527688214650117404540081771, −0.74147257859353841690614118403,
0.74147257859353841690614118403, 1.04527688214650117404540081771, 1.62458168181415894340280030754, 2.58696421747571824449021480651, 2.92305349634088367619143291018, 3.89863924625019302538537302894, 4.24201291488653967627097366766, 4.29098332677716516133742311035, 5.02125871082504210783273321456, 5.51779709043169992480590680719, 6.03776595078994905001827323637, 6.29364345372281787405357620537, 6.39723349026311220489765572830, 6.60295424472382285492880394567, 7.15523333425596544443829056455, 8.023789359013322435640585830623, 8.179772671739186686735537662119, 8.820918428669247315650459461074, 8.859469700794821283040768940734, 9.419405070665591639616921151943
