| L(s) = 1 | − 4·3-s + 8·9-s − 10·25-s − 20·27-s + 36·43-s + 12·67-s + 40·75-s + 50·81-s + 44·83-s + 24·89-s + 52·107-s + 44·121-s + 127-s − 144·129-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 8/3·9-s − 2·25-s − 3.84·27-s + 5.48·43-s + 1.46·67-s + 4.61·75-s + 50/9·81-s + 4.82·83-s + 2.54·89-s + 5.02·107-s + 4·121-s + 0.0887·127-s − 12.6·129-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 − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.457780710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457780710\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−6.96323535993980912001408834720, −6.56501129617566401945718576998, −6.15152175551460638753145340114, −6.00411614949344167019037991834, −5.98366795293723770576581380274, −5.94927020370902942254644882972, −5.91632018248476031578471745370, −5.42167068402821246228966967119, −5.03345452595656004052874214796, −4.94917232199680345463683816598, −4.75840812015071141314183607097, −4.68744777663236629113160696692, −4.02388153881232411507310705479, −4.00863332317167002079233108185, −3.79899227438733933686813987936, −3.55037195662370158802562300755, −3.39572482943623920417532168379, −2.66543913310271005286843022162, −2.45782777237964079927509745783, −2.22142644494806211316697372181, −1.80807590031542927953216856909, −1.75874994722527744116461720097, −0.823694067411163150394965490612, −0.62899967962144220574872994114, −0.57622455115973410802870284720,
0.57622455115973410802870284720, 0.62899967962144220574872994114, 0.823694067411163150394965490612, 1.75874994722527744116461720097, 1.80807590031542927953216856909, 2.22142644494806211316697372181, 2.45782777237964079927509745783, 2.66543913310271005286843022162, 3.39572482943623920417532168379, 3.55037195662370158802562300755, 3.79899227438733933686813987936, 4.00863332317167002079233108185, 4.02388153881232411507310705479, 4.68744777663236629113160696692, 4.75840812015071141314183607097, 4.94917232199680345463683816598, 5.03345452595656004052874214796, 5.42167068402821246228966967119, 5.91632018248476031578471745370, 5.94927020370902942254644882972, 5.98366795293723770576581380274, 6.00411614949344167019037991834, 6.15152175551460638753145340114, 6.56501129617566401945718576998, 6.96323535993980912001408834720
