| L(s) = 1 | − 2-s − 4-s + 5-s + 2·7-s + 8-s − 9-s − 10-s − 5·11-s − 2·14-s − 16-s − 17-s + 18-s − 19-s − 20-s + 5·22-s − 4·23-s + 25-s − 2·28-s + 8·29-s + 2·31-s + 5·32-s + 34-s + 2·35-s + 36-s − 6·37-s + 38-s + 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s − 1/3·9-s − 0.316·10-s − 1.50·11-s − 0.534·14-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 1.06·22-s − 0.834·23-s + 1/5·25-s − 0.377·28-s + 1.48·29-s + 0.359·31-s + 0.883·32-s + 0.171·34-s + 0.338·35-s + 1/6·36-s − 0.986·37-s + 0.162·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3755608648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3755608648\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 76 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−19.4858607684, −19.1169031696, −18.2862573301, −17.9899140726, −17.7402377185, −17.2306470153, −16.3871298994, −15.8436808941, −15.2908448824, −14.5014578957, −13.8984960379, −13.5197700899, −12.7835986100, −12.0643117921, −11.3139323543, −10.4886060745, −10.1604754473, −9.31611798446, −8.48193772336, −8.24162105389, −7.31047773240, −6.14898060153, −5.23755941899, −4.42861273375, −2.56912152120,
2.56912152120, 4.42861273375, 5.23755941899, 6.14898060153, 7.31047773240, 8.24162105389, 8.48193772336, 9.31611798446, 10.1604754473, 10.4886060745, 11.3139323543, 12.0643117921, 12.7835986100, 13.5197700899, 13.8984960379, 14.5014578957, 15.2908448824, 15.8436808941, 16.3871298994, 17.2306470153, 17.7402377185, 17.9899140726, 18.2862573301, 19.1169031696, 19.4858607684
