| L(s) = 1 | − 8·7-s + 12·17-s + 16·23-s − 8·25-s − 8·31-s + 4·41-s − 16·47-s + 34·49-s + 20·73-s − 24·79-s − 32·89-s + 16·97-s + 8·103-s + 32·113-s − 96·119-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 128·161-s + 163-s + 167-s − 8·169-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 2.91·17-s + 3.33·23-s − 8/5·25-s − 1.43·31-s + 0.624·41-s − 2.33·47-s + 34/7·49-s + 2.34·73-s − 2.70·79-s − 3.39·89-s + 1.62·97-s + 0.788·103-s + 3.01·113-s − 8.80·119-s + 0.909·121-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 − 10.0·161-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.744626289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.744626289\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.62737769036130342603095132388, −7.45922760667776023734989004425, −7.21039796904218709432625250389, −6.92257730599314885301537395694, −6.45108007309252564268925586539, −6.27240026633574292889261636057, −5.82372132680696849189412715924, −5.58019900362253878027437561144, −5.23724651396779942505594140063, −4.97592687656628299042940663907, −4.21206537344979041421251926597, −3.87558179656833226187493617212, −3.42688180940518004460028982928, −3.14196602028326073410732398546, −3.09409680275504381007938105393, −2.78126539175672080976280028624, −1.91720619574017406770510762004, −1.42476183201926183740417307016, −0.77523009812156767511315706229, −0.41262008320220654175248379335,
0.41262008320220654175248379335, 0.77523009812156767511315706229, 1.42476183201926183740417307016, 1.91720619574017406770510762004, 2.78126539175672080976280028624, 3.09409680275504381007938105393, 3.14196602028326073410732398546, 3.42688180940518004460028982928, 3.87558179656833226187493617212, 4.21206537344979041421251926597, 4.97592687656628299042940663907, 5.23724651396779942505594140063, 5.58019900362253878027437561144, 5.82372132680696849189412715924, 6.27240026633574292889261636057, 6.45108007309252564268925586539, 6.92257730599314885301537395694, 7.21039796904218709432625250389, 7.45922760667776023734989004425, 7.62737769036130342603095132388
