| L(s) = 1 | + 3·9-s + 8·11-s − 24·19-s − 8·29-s − 4·31-s + 19·49-s − 10·59-s − 12·61-s + 24·71-s − 36·79-s − 7·81-s − 24·89-s + 24·99-s − 46·101-s − 12·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 39·169-s − 72·171-s + ⋯ |
| L(s) = 1 | + 9-s + 2.41·11-s − 5.50·19-s − 1.48·29-s − 0.718·31-s + 19/7·49-s − 1.30·59-s − 1.53·61-s + 2.84·71-s − 4.05·79-s − 7/9·81-s − 2.54·89-s + 2.41·99-s − 4.57·101-s − 1.14·109-s − 0.363·121-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 + 0.0773·167-s + 3·169-s − 5.50·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.254450333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.254450333\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 19 T^{2} + 184 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 680 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 59 T^{2} + 1444 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 127 T^{2} + 6664 T^{4} - 127 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 107 T^{2} + 6936 T^{4} - 107 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 191 T^{2} + 14632 T^{4} - 191 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 5 T + 86 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 175 T^{2} + 15916 T^{4} - 175 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 161 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 199 T^{2} + 19840 T^{4} - 199 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 85 T^{2} + 15376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 22526 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.47720705641153756255839716199, −6.20157648963508884697311454059, −6.12349114470830002877418782494, −5.69174898879537983679204803186, −5.66242896332010733462295868651, −5.43466003893453319520443709987, −5.34506170508296469213931086513, −4.54208524705766411609994900361, −4.50418075742600502692610010654, −4.49639262794295976712704068816, −4.26527282186323017147359551419, −4.00428535033584133867341723346, −3.94674267463688639654017916825, −3.83383714250876091929595743266, −3.54952314532672067303804848579, −3.02932428616861243646637013049, −2.65485863579735171951860026419, −2.62845434297040578089079416770, −2.32894006910848013716554516161, −1.81272631460850310706014414490, −1.64604722216838827234324406590, −1.51112087306479009439007422655, −1.49522588001476367536538684544, −0.52818152986298899885087905296, −0.32703884585831648308667288333,
0.32703884585831648308667288333, 0.52818152986298899885087905296, 1.49522588001476367536538684544, 1.51112087306479009439007422655, 1.64604722216838827234324406590, 1.81272631460850310706014414490, 2.32894006910848013716554516161, 2.62845434297040578089079416770, 2.65485863579735171951860026419, 3.02932428616861243646637013049, 3.54952314532672067303804848579, 3.83383714250876091929595743266, 3.94674267463688639654017916825, 4.00428535033584133867341723346, 4.26527282186323017147359551419, 4.49639262794295976712704068816, 4.50418075742600502692610010654, 4.54208524705766411609994900361, 5.34506170508296469213931086513, 5.43466003893453319520443709987, 5.66242896332010733462295868651, 5.69174898879537983679204803186, 6.12349114470830002877418782494, 6.20157648963508884697311454059, 6.47720705641153756255839716199
