| L(s) = 1 | − 3·9-s − 12·11-s − 28·49-s + 48·59-s + 32·61-s + 24·71-s + 36·99-s + 32·109-s + 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 9-s − 3.61·11-s − 4·49-s + 6.24·59-s + 4.09·61-s + 2.84·71-s + 3.61·99-s + 3.06·109-s + 4.18·121-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 + 3.38·169-s + 0.0760·173-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 + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.438762041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438762041\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.89929429570006138288519963623, −6.68670469323539407150192765260, −6.57915888707377329483519794594, −6.51088446839144384115259268595, −5.97912177988653766011799311559, −5.55994684587194793855413809038, −5.55969291465058352616160737099, −5.50799559078641462478149683067, −5.08948486018141563716444767019, −5.06966633744549355466352372895, −4.98534942806625248997553058568, −4.59021427995216149536628631711, −4.15376894899442039769270520641, −3.88497527519004272860571672959, −3.57142530586338997236281218908, −3.53578843283938185265770384955, −3.11254192744972774004589500809, −2.74845397359810433691047741210, −2.58563794369830486514893799642, −2.35813636731641701614943154646, −2.13322853984231446408569863864, −1.93729231933103658293255531009, −1.15855995616410856790063049314, −0.57839160716845157709482419622, −0.40530332179115754764925378030,
0.40530332179115754764925378030, 0.57839160716845157709482419622, 1.15855995616410856790063049314, 1.93729231933103658293255531009, 2.13322853984231446408569863864, 2.35813636731641701614943154646, 2.58563794369830486514893799642, 2.74845397359810433691047741210, 3.11254192744972774004589500809, 3.53578843283938185265770384955, 3.57142530586338997236281218908, 3.88497527519004272860571672959, 4.15376894899442039769270520641, 4.59021427995216149536628631711, 4.98534942806625248997553058568, 5.06966633744549355466352372895, 5.08948486018141563716444767019, 5.50799559078641462478149683067, 5.55969291465058352616160737099, 5.55994684587194793855413809038, 5.97912177988653766011799311559, 6.51088446839144384115259268595, 6.57915888707377329483519794594, 6.68670469323539407150192765260, 6.89929429570006138288519963623
