| L(s) = 1 | + 4·7-s − 2·25-s + 20·31-s − 2·49-s − 28·73-s + 20·79-s + 4·97-s + 28·103-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s − 8·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 2/5·25-s + 3.59·31-s − 2/7·49-s − 3.27·73-s + 2.25·79-s + 0.406·97-s + 2.75·103-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 + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s − 0.604·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.325806782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.325806782\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.136398946569171392790795517249, −8.618850349946029972549487438837, −8.361255339148681707197923814251, −8.190317218400559501156480400180, −7.65257603976594975823956538711, −7.44896606746174051922701192764, −6.91454146600431427713492292885, −6.36523513407901939637797751002, −6.13700566218195458354613235626, −5.72841593976055951052620493493, −4.99907292809383628661532444156, −4.88450018484767907656374024991, −4.30943750807639038667710226447, −4.29753523993744654552708635154, −3.20401742467794596651434910843, −3.14374695409377479797205561887, −2.30212776630714975831747505426, −1.92621689418655432325939293744, −1.26177245514393104920732673595, −0.68443941890410295313288140747,
0.68443941890410295313288140747, 1.26177245514393104920732673595, 1.92621689418655432325939293744, 2.30212776630714975831747505426, 3.14374695409377479797205561887, 3.20401742467794596651434910843, 4.29753523993744654552708635154, 4.30943750807639038667710226447, 4.88450018484767907656374024991, 4.99907292809383628661532444156, 5.72841593976055951052620493493, 6.13700566218195458354613235626, 6.36523513407901939637797751002, 6.91454146600431427713492292885, 7.44896606746174051922701192764, 7.65257603976594975823956538711, 8.190317218400559501156480400180, 8.361255339148681707197923814251, 8.618850349946029972549487438837, 9.136398946569171392790795517249
