| L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s − 3·9-s + 4·11-s + 2·13-s + 4·15-s + 2·17-s − 8·19-s + 4·21-s + 12·23-s + 25-s + 14·27-s − 12·29-s − 8·33-s + 4·35-s − 2·37-s − 4·39-s − 6·43-s + 6·45-s + 6·47-s − 3·49-s − 4·51-s − 12·53-s − 8·55-s + 16·57-s − 16·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s − 9-s + 1.20·11-s + 0.554·13-s + 1.03·15-s + 0.485·17-s − 1.83·19-s + 0.872·21-s + 2.50·23-s + 1/5·25-s + 2.69·27-s − 2.22·29-s − 1.39·33-s + 0.676·35-s − 0.328·37-s − 0.640·39-s − 0.914·43-s + 0.894·45-s + 0.875·47-s − 3/7·49-s − 0.560·51-s − 1.64·53-s − 1.07·55-s + 2.11·57-s − 2.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_4$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 106 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 26 T + 303 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.364228814592878184235097095689, −8.253015343179750951591283580025, −7.49366784478139290858999077595, −7.43002577708871257783988332091, −6.68744252414480750951189595635, −6.48427800957551667537928009509, −6.17696813675356189217766874794, −6.09152647687920253571460189719, −5.25740146196813861050792486180, −5.10684565461629665573783564608, −4.75753706777823375618693944505, −4.09677600755433771275645233741, −3.60508235626891630195647273566, −3.45856098478754736423696575872, −2.95564708280864717832401233156, −2.38018396635219455012851617521, −1.54524151364170484780451807308, −1.00843164166974151095249506504, 0, 0,
1.00843164166974151095249506504, 1.54524151364170484780451807308, 2.38018396635219455012851617521, 2.95564708280864717832401233156, 3.45856098478754736423696575872, 3.60508235626891630195647273566, 4.09677600755433771275645233741, 4.75753706777823375618693944505, 5.10684565461629665573783564608, 5.25740146196813861050792486180, 6.09152647687920253571460189719, 6.17696813675356189217766874794, 6.48427800957551667537928009509, 6.68744252414480750951189595635, 7.43002577708871257783988332091, 7.49366784478139290858999077595, 8.253015343179750951591283580025, 8.364228814592878184235097095689
