| L(s) = 1 | − 6·11-s + 8·13-s − 2·25-s + 4·37-s − 24·47-s + 2·49-s − 30·59-s + 16·61-s + 12·71-s − 22·73-s + 24·83-s + 26·97-s − 6·107-s + 8·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s − 48·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.80·11-s + 2.21·13-s − 2/5·25-s + 0.657·37-s − 3.50·47-s + 2/7·49-s − 3.90·59-s + 2.04·61-s + 1.42·71-s − 2.57·73-s + 2.63·83-s + 2.63·97-s − 0.580·107-s + 0.766·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.649587999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649587999\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.12270157187922772990330834902, −9.244079133681734161641388893418, −9.211201324844553621160929352931, −8.476238255088389013042723931012, −8.247911169354740540091454800156, −7.83785219100956297340427987029, −7.67348591381124405402416316808, −6.96380024792757083150558977815, −6.40614632488096096568759833373, −6.11964100201384785641236438444, −5.84027167838508301485762090286, −5.09619397639961647996939544456, −4.91675999308331893814742871519, −4.31894829767982971077675507345, −3.65090867216549021549229303654, −3.24939134565181025566404899769, −2.87389874500256481262357676882, −1.99848449014906957936616513231, −1.53795661505061321203269793216, −0.54148550583982276674566827524,
0.54148550583982276674566827524, 1.53795661505061321203269793216, 1.99848449014906957936616513231, 2.87389874500256481262357676882, 3.24939134565181025566404899769, 3.65090867216549021549229303654, 4.31894829767982971077675507345, 4.91675999308331893814742871519, 5.09619397639961647996939544456, 5.84027167838508301485762090286, 6.11964100201384785641236438444, 6.40614632488096096568759833373, 6.96380024792757083150558977815, 7.67348591381124405402416316808, 7.83785219100956297340427987029, 8.247911169354740540091454800156, 8.476238255088389013042723931012, 9.211201324844553621160929352931, 9.244079133681734161641388893418, 10.12270157187922772990330834902
