| L(s) = 1 | − 3-s − 7-s − 9-s + 4·11-s + 3·13-s − 17-s + 12·19-s + 21-s − 2·23-s + 4·29-s − 2·31-s − 4·33-s + 5·37-s − 3·39-s + 3·47-s − 9·49-s + 51-s + 5·53-s − 12·57-s + 5·59-s − 6·61-s + 63-s + 13·67-s + 2·69-s + 12·71-s + 13·73-s − 4·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 1/3·9-s + 1.20·11-s + 0.832·13-s − 0.242·17-s + 2.75·19-s + 0.218·21-s − 0.417·23-s + 0.742·29-s − 0.359·31-s − 0.696·33-s + 0.821·37-s − 0.480·39-s + 0.437·47-s − 9/7·49-s + 0.140·51-s + 0.686·53-s − 1.58·57-s + 0.650·59-s − 0.768·61-s + 0.125·63-s + 1.58·67-s + 0.240·69-s + 1.42·71-s + 1.52·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.585330753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.585330753\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 86 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 13 T + 172 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 161 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 13 T + 184 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−9.172762470470051490444627615754, −9.092076923693618715874642940967, −8.293517805648658975711035925374, −8.106213529203737357872136755797, −7.65897197465997298265102625966, −7.26848940680705002312890243293, −6.66966261268072044245020596963, −6.48896516580428713534902439283, −6.20506486234341460071425962328, −5.56084841742031625809507161891, −5.38153171828906561416909522475, −4.96118960805806287013027839377, −4.36457078517489556429904599176, −3.83613917597980797291098993997, −3.40578182087459645642737418912, −3.23633670788436589639173143805, −2.41500774820306958132443642940, −1.82893913297148645715508309697, −0.968718800121514067773161276064, −0.76389911462393003627685656846,
0.76389911462393003627685656846, 0.968718800121514067773161276064, 1.82893913297148645715508309697, 2.41500774820306958132443642940, 3.23633670788436589639173143805, 3.40578182087459645642737418912, 3.83613917597980797291098993997, 4.36457078517489556429904599176, 4.96118960805806287013027839377, 5.38153171828906561416909522475, 5.56084841742031625809507161891, 6.20506486234341460071425962328, 6.48896516580428713534902439283, 6.66966261268072044245020596963, 7.26848940680705002312890243293, 7.65897197465997298265102625966, 8.106213529203737357872136755797, 8.293517805648658975711035925374, 9.092076923693618715874642940967, 9.172762470470051490444627615754
