| L(s) = 1 | − 4·3-s + 4-s − 3·7-s + 6·9-s + 6·11-s − 4·12-s − 2·13-s + 16-s + 5·17-s − 2·19-s + 12·21-s − 10·25-s + 4·27-s − 3·28-s − 6·29-s − 8·31-s − 24·33-s + 6·36-s − 2·37-s + 8·39-s + 12·41-s + 16·43-s + 6·44-s − 12·47-s − 4·48-s + 6·49-s − 20·51-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1/2·4-s − 1.13·7-s + 2·9-s + 1.80·11-s − 1.15·12-s − 0.554·13-s + 1/4·16-s + 1.21·17-s − 0.458·19-s + 2.61·21-s − 2·25-s + 0.769·27-s − 0.566·28-s − 1.11·29-s − 1.43·31-s − 4.17·33-s + 36-s − 0.328·37-s + 1.28·39-s + 1.87·41-s + 2.43·43-s + 0.904·44-s − 1.75·47-s − 0.577·48-s + 6/7·49-s − 2.80·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2474290657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2474290657\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.9292301410, −19.5800270198, −19.1494492613, −18.2120541317, −17.4469801976, −17.2128532162, −16.6317948443, −16.3370810014, −15.7312805996, −14.6077730657, −14.2197849424, −12.8595646329, −12.3052609901, −11.9583441737, −11.2313614144, −10.8489466872, −9.76554711946, −9.28607198826, −7.57571100089, −6.65330731261, −5.99911855172, −5.57928681743, −3.90229547123,
3.90229547123, 5.57928681743, 5.99911855172, 6.65330731261, 7.57571100089, 9.28607198826, 9.76554711946, 10.8489466872, 11.2313614144, 11.9583441737, 12.3052609901, 12.8595646329, 14.2197849424, 14.6077730657, 15.7312805996, 16.3370810014, 16.6317948443, 17.2128532162, 17.4469801976, 18.2120541317, 19.1494492613, 19.5800270198, 19.9292301410
