L(s) = 1 | + 4-s − 6·11-s + 4·19-s − 12·29-s + 8·31-s + 18·41-s − 6·44-s − 10·49-s − 6·59-s − 16·61-s − 64-s + 48·71-s + 4·76-s − 8·79-s + 24·89-s + 64·109-s − 12·116-s + 31·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 1.80·11-s + 0.917·19-s − 2.22·29-s + 1.43·31-s + 2.81·41-s − 0.904·44-s − 1.42·49-s − 0.781·59-s − 2.04·61-s − 1/8·64-s + 5.69·71-s + 0.458·76-s − 0.900·79-s + 2.54·89-s + 6.13·109-s − 1.11·116-s + 2.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.894936300\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.894936300\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 22 T^{2} - 6405 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.64171551380840074625874180010, −6.57984897491581718717752985482, −6.57772348216509778509032095579, −6.22495984260923507135459695052, −5.86502553680360006878396811997, −5.79477951497956038705936519320, −5.63260456470012453038019217777, −5.29883376164088731735363203912, −5.10482273036100211031853462007, −4.85728071155618079031030271883, −4.64808054925628873923713421850, −4.53322895786870081684100637872, −4.02510295095209927707796648732, −3.99437028918383817442317007756, −3.46177340284659046461767636430, −3.27270078000422020160256890644, −3.06664622790837124749870516892, −3.03267832259356756113865106783, −2.33896801028541606784872011459, −2.25854854560442798381068123367, −2.07486634272720704910137362105, −1.82342486060703456081694315191, −1.12354455649615385936801076877, −0.809417021435246842165111118282, −0.40386496028194583567454029833,
0.40386496028194583567454029833, 0.809417021435246842165111118282, 1.12354455649615385936801076877, 1.82342486060703456081694315191, 2.07486634272720704910137362105, 2.25854854560442798381068123367, 2.33896801028541606784872011459, 3.03267832259356756113865106783, 3.06664622790837124749870516892, 3.27270078000422020160256890644, 3.46177340284659046461767636430, 3.99437028918383817442317007756, 4.02510295095209927707796648732, 4.53322895786870081684100637872, 4.64808054925628873923713421850, 4.85728071155618079031030271883, 5.10482273036100211031853462007, 5.29883376164088731735363203912, 5.63260456470012453038019217777, 5.79477951497956038705936519320, 5.86502553680360006878396811997, 6.22495984260923507135459695052, 6.57772348216509778509032095579, 6.57984897491581718717752985482, 6.64171551380840074625874180010