| L(s) = 1 | − 4·7-s − 26·19-s − 50·25-s − 44·43-s − 86·49-s − 148·61-s − 92·73-s − 242·121-s + 127-s + 131-s + 104·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 146·169-s + 173-s + 200·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4/7·7-s − 1.36·19-s − 2·25-s − 1.02·43-s − 1.75·49-s − 2.42·61-s − 1.26·73-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.781·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.863·169-s + 0.00578·173-s + 8/7·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5579427355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5579427355\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 26 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 122 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
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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
−10.55882131995040134463913996220, −10.00229317590574280893089601781, −9.703544158313257447368248180357, −9.169612419411650410219666604447, −8.918602733232598266644600421781, −8.106805326885983890349195914311, −8.074341752095899993765634886160, −7.49226122326568018692123791774, −6.90483454677010773843409982406, −6.34757415330698108388573505031, −6.22494029760490995195735330313, −5.59418225127288183103015524993, −5.06501410485378818501030534332, −4.29757220011257812927228317226, −4.15052676258395661547726760607, −3.29908411790347615167131401283, −2.94008588727046206248111386239, −1.99768614249191716496183668197, −1.60954175719711653424967202922, −0.25647752265183202834347007699,
0.25647752265183202834347007699, 1.60954175719711653424967202922, 1.99768614249191716496183668197, 2.94008588727046206248111386239, 3.29908411790347615167131401283, 4.15052676258395661547726760607, 4.29757220011257812927228317226, 5.06501410485378818501030534332, 5.59418225127288183103015524993, 6.22494029760490995195735330313, 6.34757415330698108388573505031, 6.90483454677010773843409982406, 7.49226122326568018692123791774, 8.074341752095899993765634886160, 8.106805326885983890349195914311, 8.918602733232598266644600421781, 9.169612419411650410219666604447, 9.703544158313257447368248180357, 10.00229317590574280893089601781, 10.55882131995040134463913996220
