| L(s) = 1 | + 5-s − 2·7-s − 9-s + 2·11-s + 2·13-s + 19-s + 23-s − 2·35-s + 2·37-s + 41-s − 45-s − 2·47-s + 49-s − 53-s + 2·55-s + 2·59-s + 2·63-s + 2·65-s − 4·77-s + 89-s − 4·91-s + 95-s − 2·99-s − 2·103-s + 115-s − 2·117-s + 121-s + ⋯ |
| L(s) = 1 | + 5-s − 2·7-s − 9-s + 2·11-s + 2·13-s + 19-s + 23-s − 2·35-s + 2·37-s + 41-s − 45-s − 2·47-s + 49-s − 53-s + 2·55-s + 2·59-s + 2·63-s + 2·65-s − 4·77-s + 89-s − 4·91-s + 95-s − 2·99-s − 2·103-s + 115-s − 2·117-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.683173454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683173454\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.085091548687465994327044850140, −9.000732084135495242370467687617, −8.377116103663000337426178673480, −8.142584297363673569685049422218, −7.58042657672888258654992696404, −6.93721810886196574099493167579, −6.56065178387043791894861137201, −6.49728019716131598792827613114, −6.07407167079660713860752937162, −5.91933392737660480258898665736, −5.45966493862891719914594698633, −4.96400710779247508866763562378, −4.17343833389616459285210140999, −3.92775369384416416270841555023, −3.41712815029325000519048335661, −3.09414581716443576160011863329, −2.83135266384425356451791987173, −2.02194835953953711447701553187, −1.29828814189414780789059437273, −0.937359203617292703491779511416,
0.937359203617292703491779511416, 1.29828814189414780789059437273, 2.02194835953953711447701553187, 2.83135266384425356451791987173, 3.09414581716443576160011863329, 3.41712815029325000519048335661, 3.92775369384416416270841555023, 4.17343833389616459285210140999, 4.96400710779247508866763562378, 5.45966493862891719914594698633, 5.91933392737660480258898665736, 6.07407167079660713860752937162, 6.49728019716131598792827613114, 6.56065178387043791894861137201, 6.93721810886196574099493167579, 7.58042657672888258654992696404, 8.142584297363673569685049422218, 8.377116103663000337426178673480, 9.000732084135495242370467687617, 9.085091548687465994327044850140
