| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s − 3·8-s + 3·9-s + 10-s − 4·11-s + 2·12-s − 4·13-s − 2·15-s + 16-s − 11·17-s − 3·18-s + 7·19-s − 20-s + 4·22-s + 2·23-s − 6·24-s − 5·25-s + 4·26-s + 4·27-s − 10·29-s + 2·30-s − 31-s + 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 1.06·8-s + 9-s + 0.316·10-s − 1.20·11-s + 0.577·12-s − 1.10·13-s − 0.516·15-s + 1/4·16-s − 2.66·17-s − 0.707·18-s + 1.60·19-s − 0.223·20-s + 0.852·22-s + 0.417·23-s − 1.22·24-s − 25-s + 0.784·26-s + 0.769·27-s − 1.85·29-s + 0.365·30-s − 0.179·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11431161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11431161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 90 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 56 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 217 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 20 T + 229 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 210 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 230 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.294850185482275641643730706433, −8.251726715861770275855499810372, −7.69584217859276073660478783203, −7.32653951904363847218374502153, −7.26062013500456264763020107390, −6.66099184086376268156908147175, −6.47890841020075686220457964953, −5.77225578078412080225142227353, −5.32186511420367947734616364083, −5.01681873223129757355757135019, −4.51345740386899872100072446248, −4.06175623241001680708149113304, −3.55417765653414788827892534489, −3.20538872994800615734939213847, −2.51623208772260485246108126343, −2.30947267854393450618928626719, −2.13855963841063236199853621938, −1.17225012728045377499361094155, 0, 0,
1.17225012728045377499361094155, 2.13855963841063236199853621938, 2.30947267854393450618928626719, 2.51623208772260485246108126343, 3.20538872994800615734939213847, 3.55417765653414788827892534489, 4.06175623241001680708149113304, 4.51345740386899872100072446248, 5.01681873223129757355757135019, 5.32186511420367947734616364083, 5.77225578078412080225142227353, 6.47890841020075686220457964953, 6.66099184086376268156908147175, 7.26062013500456264763020107390, 7.32653951904363847218374502153, 7.69584217859276073660478783203, 8.251726715861770275855499810372, 8.294850185482275641643730706433
