| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s − 3·7-s + 4·9-s + 6·10-s − 7·11-s − 3·12-s − 13-s + 6·14-s + 9·15-s + 16-s − 7·17-s − 8·18-s − 9·19-s − 3·20-s + 9·21-s + 14·22-s − 4·23-s + 25-s + 2·26-s − 6·27-s − 3·28-s − 3·29-s − 18·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s − 1.13·7-s + 4/3·9-s + 1.89·10-s − 2.11·11-s − 0.866·12-s − 0.277·13-s + 1.60·14-s + 2.32·15-s + 1/4·16-s − 1.69·17-s − 1.88·18-s − 2.06·19-s − 0.670·20-s + 1.96·21-s + 2.98·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s − 1.15·27-s − 0.566·28-s − 0.557·29-s − 3.28·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206111 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206111 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 2609 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 65 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 38 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 50 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 106 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T + 82 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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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
−13.7033230898, −13.2831178543, −13.1285700839, −12.6297126310, −12.1744855241, −11.7927743579, −11.4125961333, −10.8702781873, −10.6717242145, −10.2110599414, −9.93078435302, −9.34454256837, −8.72745453441, −8.36065480700, −8.02687635305, −7.50019808099, −6.95010449263, −6.36696733419, −6.17396318055, −5.41027297756, −4.93004199175, −4.28623702265, −3.82254947249, −2.82375788484, −2.06834149820, 0, 0, 0,
2.06834149820, 2.82375788484, 3.82254947249, 4.28623702265, 4.93004199175, 5.41027297756, 6.17396318055, 6.36696733419, 6.95010449263, 7.50019808099, 8.02687635305, 8.36065480700, 8.72745453441, 9.34454256837, 9.93078435302, 10.2110599414, 10.6717242145, 10.8702781873, 11.4125961333, 11.7927743579, 12.1744855241, 12.6297126310, 13.1285700839, 13.2831178543, 13.7033230898
