| L(s) = 1 | + 2·2-s + 2·4-s − 6·7-s + 2·9-s − 12·14-s − 4·16-s + 4·17-s + 4·18-s + 8·23-s − 6·25-s − 12·28-s − 10·31-s − 8·32-s + 8·34-s + 4·36-s + 8·41-s + 16·46-s + 10·47-s + 14·49-s − 12·50-s − 20·62-s − 12·63-s − 8·64-s + 8·68-s + 12·71-s + 5·73-s + 8·79-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 2.26·7-s + 2/3·9-s − 3.20·14-s − 16-s + 0.970·17-s + 0.942·18-s + 1.66·23-s − 6/5·25-s − 2.26·28-s − 1.79·31-s − 1.41·32-s + 1.37·34-s + 2/3·36-s + 1.24·41-s + 2.35·46-s + 1.45·47-s + 2·49-s − 1.69·50-s − 2.54·62-s − 1.51·63-s − 64-s + 0.970·68-s + 1.42·71-s + 0.585·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.342826702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.342826702\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−12.63974512870399257039650739060, −12.16551706224909237874193464731, −11.28268555030511406812863015604, −10.69724195694489126020802959948, −9.898328950839830403199017345697, −9.375040266738990391293249866098, −9.088421462106022162612758583749, −7.72922427819488925640567546189, −7.02428954566466171273305250222, −6.56249730409478760856133043539, −5.79621689921985941311583977855, −5.28234939584660765761721801828, −4.02676825708374336048445873000, −3.55176454243207444733428451920, −2.72454121493799864819769185152,
2.72454121493799864819769185152, 3.55176454243207444733428451920, 4.02676825708374336048445873000, 5.28234939584660765761721801828, 5.79621689921985941311583977855, 6.56249730409478760856133043539, 7.02428954566466171273305250222, 7.72922427819488925640567546189, 9.088421462106022162612758583749, 9.375040266738990391293249866098, 9.898328950839830403199017345697, 10.69724195694489126020802959948, 11.28268555030511406812863015604, 12.16551706224909237874193464731, 12.63974512870399257039650739060
