| L(s) = 1 | + 3-s − 5-s − 2·9-s + 8·11-s + 2·13-s − 15-s + 2·17-s − 10·19-s + 8·23-s − 6·25-s − 2·27-s + 4·29-s − 11·31-s + 8·33-s − 2·37-s + 2·39-s + 5·41-s − 5·43-s + 2·45-s − 2·47-s + 2·51-s + 5·53-s − 8·55-s − 10·57-s + 16·59-s − 13·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s + 2.41·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 2.29·19-s + 1.66·23-s − 6/5·25-s − 0.384·27-s + 0.742·29-s − 1.97·31-s + 1.39·33-s − 0.328·37-s + 0.320·39-s + 0.780·41-s − 0.762·43-s + 0.298·45-s − 0.291·47-s + 0.280·51-s + 0.686·53-s − 1.07·55-s − 1.32·57-s + 2.08·59-s − 1.66·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.797155884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.797155884\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 11 T + 89 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 7 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_4$ | \( 1 + 5 T + 89 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 109 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 83 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 67 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 146 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 23 T + 323 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
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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
−8.718648842119222093580900860835, −8.671585257688785724370683754813, −8.147484629416981854397868463540, −7.82180971822947860581144124178, −7.21640803018341674063951741026, −7.00439962645016534362261510177, −6.54119028399847261641829827437, −6.28534450421200033161846546478, −5.91757040486787312808952087741, −5.46343236223243984320269451537, −4.92662995774910069955976555214, −4.41241946058535657515881015690, −3.91523616946610771549225298010, −3.87696307826549989272217908119, −3.32039345517112502669803111116, −2.97928863720325613192907106650, −2.11828384774845839492676935403, −1.90459434540108135576791933344, −1.21959480004910906126175542136, −0.51938492046778447778210551007,
0.51938492046778447778210551007, 1.21959480004910906126175542136, 1.90459434540108135576791933344, 2.11828384774845839492676935403, 2.97928863720325613192907106650, 3.32039345517112502669803111116, 3.87696307826549989272217908119, 3.91523616946610771549225298010, 4.41241946058535657515881015690, 4.92662995774910069955976555214, 5.46343236223243984320269451537, 5.91757040486787312808952087741, 6.28534450421200033161846546478, 6.54119028399847261641829827437, 7.00439962645016534362261510177, 7.21640803018341674063951741026, 7.82180971822947860581144124178, 8.147484629416981854397868463540, 8.671585257688785724370683754813, 8.718648842119222093580900860835
