L(s) = 1 | − 6·9-s − 12·13-s − 6·25-s + 20·29-s − 14·49-s + 20·61-s + 27·81-s + 20·89-s + 36·97-s + 4·101-s − 12·109-s − 28·113-s + 72·117-s − 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 2·9-s − 3.32·13-s − 6/5·25-s + 3.71·29-s − 2·49-s + 2.56·61-s + 3·81-s + 2.11·89-s + 3.65·97-s + 0.398·101-s − 1.14·109-s − 2.63·113-s + 6.65·117-s − 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 + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7328972707\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7328972707\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−9.046460458387871865172788417874, −8.388528685678114329881751537871, −7.86574055998535746161422058704, −7.84811471855830963136818273258, −7.02884226586020397437679457068, −6.42244536959809046172401939908, −6.27282080646876620831649653268, −5.28260967175901923444909960051, −5.01452297596172644651029655270, −4.79895961140065649382690544917, −3.84502571910183330789299225162, −2.94636053881520680537966486208, −2.63895104553179739845027310173, −2.18612728567174242201060717336, −0.48307566503461853621511078883,
0.48307566503461853621511078883, 2.18612728567174242201060717336, 2.63895104553179739845027310173, 2.94636053881520680537966486208, 3.84502571910183330789299225162, 4.79895961140065649382690544917, 5.01452297596172644651029655270, 5.28260967175901923444909960051, 6.27282080646876620831649653268, 6.42244536959809046172401939908, 7.02884226586020397437679457068, 7.84811471855830963136818273258, 7.86574055998535746161422058704, 8.388528685678114329881751537871, 9.046460458387871865172788417874