| L(s) = 1 | + 46·9-s − 180·17-s + 250·25-s + 1.04e3·41-s − 686·49-s + 860·73-s + 1.38e3·81-s − 2.05e3·89-s + 3.82e3·97-s − 540·113-s − 2.33e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8.28e3·153-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1.70·9-s − 2.56·17-s + 2·25-s + 3.97·41-s − 2·49-s + 1.37·73-s + 1.90·81-s − 2.44·89-s + 3.99·97-s − 0.449·113-s − 1.75·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 4.37·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.287645930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287645930\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )( 1 + 10 T + p^{3} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 18 T + p^{3} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )( 1 + 106 T + p^{3} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 522 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 290 T + p^{3} T^{2} )( 1 + 290 T + p^{3} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 846 T + p^{3} T^{2} )( 1 + 846 T + p^{3} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )( 1 + 70 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 430 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 1350 T + p^{3} T^{2} )( 1 + 1350 T + p^{3} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 1026 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1910 T + p^{3} T^{2} )^{2} \) |
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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
−12.94212640267255725999332712455, −12.86613391569272739481506734153, −12.32145352280495509802959818033, −11.42036153002511934762883501937, −10.84174934320969070722998291614, −10.83996791860317420113309508985, −9.996913186961124265760855940422, −9.267765683929770566087148110692, −9.162901398394641071451612150416, −8.359883756521301568888731607607, −7.65244323634178350286896130015, −7.05336301730707782102125372761, −6.64124692147669390435640856039, −6.10170787573384112802259291180, −4.91964601291452429483575426540, −4.50670355090226620700501858155, −4.01134026870600113444691267406, −2.79117855394793387110811803056, −1.94800741825499369940914819677, −0.819017669647580252860004126787,
0.819017669647580252860004126787, 1.94800741825499369940914819677, 2.79117855394793387110811803056, 4.01134026870600113444691267406, 4.50670355090226620700501858155, 4.91964601291452429483575426540, 6.10170787573384112802259291180, 6.64124692147669390435640856039, 7.05336301730707782102125372761, 7.65244323634178350286896130015, 8.359883756521301568888731607607, 9.162901398394641071451612150416, 9.267765683929770566087148110692, 9.996913186961124265760855940422, 10.83996791860317420113309508985, 10.84174934320969070722998291614, 11.42036153002511934762883501937, 12.32145352280495509802959818033, 12.86613391569272739481506734153, 12.94212640267255725999332712455
