| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 4·7-s + 8-s + 9-s − 2·11-s + 2·12-s − 4·14-s + 16-s − 2·17-s + 18-s + 6·19-s − 8·21-s − 2·22-s − 4·23-s + 2·24-s + 2·27-s − 4·28-s − 8·29-s + 12·31-s + 32-s − 4·33-s − 2·34-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.37·19-s − 1.74·21-s − 0.426·22-s − 0.834·23-s + 0.408·24-s + 0.384·27-s − 0.755·28-s − 1.48·29-s + 2.15·31-s + 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.730017585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730017585\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−16.2729884099, −15.8558735832, −15.7757131397, −15.0468991436, −14.3890381327, −14.2754339531, −13.3398538128, −13.3181822032, −12.9167058262, −12.1846203232, −11.5780567836, −11.1600742237, −10.0224451499, −9.93072373032, −9.44701579288, −8.56747242148, −8.07172921856, −7.52156329683, −6.57858828761, −6.35306147257, −5.34945783065, −4.60235027296, −3.34359627566, −3.29699975126, −2.26917003136,
2.26917003136, 3.29699975126, 3.34359627566, 4.60235027296, 5.34945783065, 6.35306147257, 6.57858828761, 7.52156329683, 8.07172921856, 8.56747242148, 9.44701579288, 9.93072373032, 10.0224451499, 11.1600742237, 11.5780567836, 12.1846203232, 12.9167058262, 13.3181822032, 13.3398538128, 14.2754339531, 14.3890381327, 15.0468991436, 15.7757131397, 15.8558735832, 16.2729884099
