| L(s) = 1 | − 14·5-s − 18·9-s + 24·13-s + 98·25-s + 80·29-s − 46·37-s − 98·41-s + 252·45-s − 180·53-s + 44·61-s − 336·65-s − 206·73-s + 243·81-s − 82·89-s + 274·97-s − 62·109-s − 448·113-s − 432·117-s − 350·125-s + 127-s + 131-s + 137-s + 139-s − 1.12e3·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2.79·5-s − 2·9-s + 1.84·13-s + 3.91·25-s + 2.75·29-s − 1.24·37-s − 2.39·41-s + 28/5·45-s − 3.39·53-s + 0.721·61-s − 5.16·65-s − 2.82·73-s + 3·81-s − 0.921·89-s + 2.82·97-s − 0.568·109-s − 3.96·113-s − 3.69·117-s − 2.79·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.72·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2520632553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2520632553\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 24 T + p^{2} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )( 1 + 8 T + p^{2} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )( 1 + 80 T + p^{2} T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )( 1 + 160 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )( 1 - 130 T + p^{2} T^{2} ) \) |
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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
−11.41012445304543214122650028070, −10.98801662990072814601146941475, −10.52838748846064366779925804968, −10.08492185490885724146589830752, −8.998518068969086855493034632019, −8.697190444327680956824538648298, −8.469057703608157818500561116102, −7.956044444775796745770994872324, −7.955303076044463993183207242137, −7.04528389378919521813763069367, −6.48223321822479379400090335885, −6.22543251224299680260777824096, −5.35306127225151793049558822652, −4.79239182285770299085814992403, −4.28761243844372223322587229404, −3.43881269410762970536797088313, −3.40333079498714128439871200488, −2.84690447047710187454759625908, −1.31873714462970097180279669404, −0.23461819360420644172229273626,
0.23461819360420644172229273626, 1.31873714462970097180279669404, 2.84690447047710187454759625908, 3.40333079498714128439871200488, 3.43881269410762970536797088313, 4.28761243844372223322587229404, 4.79239182285770299085814992403, 5.35306127225151793049558822652, 6.22543251224299680260777824096, 6.48223321822479379400090335885, 7.04528389378919521813763069367, 7.955303076044463993183207242137, 7.956044444775796745770994872324, 8.469057703608157818500561116102, 8.697190444327680956824538648298, 8.998518068969086855493034632019, 10.08492185490885724146589830752, 10.52838748846064366779925804968, 10.98801662990072814601146941475, 11.41012445304543214122650028070
