| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·5-s − 2·6-s + 3·7-s − 3·8-s − 2·9-s + 2·10-s + 2·11-s + 2·12-s − 3·14-s − 4·15-s + 16-s + 17-s + 2·18-s + 19-s − 2·20-s + 6·21-s − 2·22-s − 5·23-s − 6·24-s + 2·25-s − 10·27-s + 3·28-s − 5·29-s + 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s + 1.13·7-s − 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.603·11-s + 0.577·12-s − 0.801·14-s − 1.03·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 0.229·19-s − 0.447·20-s + 1.30·21-s − 0.426·22-s − 1.04·23-s − 1.22·24-s + 2/5·25-s − 1.92·27-s + 0.566·28-s − 0.928·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3349 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3349 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6810411511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6810411511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 197 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 18 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + T + 64 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 146 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−18.1046582305, −17.6539911456, −17.0588307481, −16.6856681416, −15.7259659146, −15.5240416373, −14.8035245868, −14.5119862249, −14.0739411756, −13.5099716515, −12.4555523632, −11.8532895631, −11.5393004146, −11.0870978899, −10.2046327229, −9.29748754565, −8.89244456852, −8.44835817021, −7.82458356398, −7.44913939960, −6.29961221708, −5.48692681762, −4.20385690206, −3.32688345817, −2.26865064540,
2.26865064540, 3.32688345817, 4.20385690206, 5.48692681762, 6.29961221708, 7.44913939960, 7.82458356398, 8.44835817021, 8.89244456852, 9.29748754565, 10.2046327229, 11.0870978899, 11.5393004146, 11.8532895631, 12.4555523632, 13.5099716515, 14.0739411756, 14.5119862249, 14.8035245868, 15.5240416373, 15.7259659146, 16.6856681416, 17.0588307481, 17.6539911456, 18.1046582305
