| L(s) = 1 | − 176·7-s + 342·9-s + 1.18e3·17-s − 8.20e3·23-s + 3.33e3·25-s − 8.51e3·31-s − 3.44e4·41-s + 2.59e3·47-s − 1.03e4·49-s − 6.01e4·63-s − 9.37e4·71-s − 1.35e5·73-s + 1.53e5·79-s + 5.79e4·81-s − 5.95e4·89-s − 2.44e5·97-s − 5.45e4·103-s − 5.92e4·113-s − 2.09e5·119-s + 3.05e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4.06e5·153-s + ⋯ |
| L(s) = 1 | − 1.35·7-s + 1.40·9-s + 0.996·17-s − 3.23·23-s + 1.06·25-s − 1.59·31-s − 3.20·41-s + 0.171·47-s − 0.617·49-s − 1.91·63-s − 2.20·71-s − 2.96·73-s + 2.77·79-s + 0.980·81-s − 0.796·89-s − 2.64·97-s − 0.506·103-s − 0.436·113-s − 1.35·119-s + 0.189·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 1.40·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1870435552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1870435552\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 - 38 p^{2} T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3334 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 88 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 30502 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 567862 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 594 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 11782 p^{2} T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4104 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 40669462 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4256 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138599110 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 17226 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 147606886 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 1296 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 456374950 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1371050374 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 482463958 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2224486870 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 46872 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 67562 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 76912 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3292624630 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 29754 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 122398 T + p^{5} T^{2} )^{2} \) |
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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
−11.60541226325730901937735502807, −10.55201879853308613260439031575, −10.42232115203152666088650875087, −9.869342503231410947193766446648, −9.752167845702409155893436600803, −9.126851443848936814166943575674, −8.444285807094076060195889002243, −7.939785486910255471890727563564, −7.42877401219022593199367331278, −6.81986702064647802816185123443, −6.54594806974167425351621693991, −5.85997406064744443370246341093, −5.38101408225905076987449756363, −4.56262939147201569380288848886, −3.92466635453073841614723316181, −3.51872166651258097621825353372, −2.88284002834170367663423348140, −1.76578556463569406586695507691, −1.47415393542951126521269706345, −0.11909709289723749298048491877,
0.11909709289723749298048491877, 1.47415393542951126521269706345, 1.76578556463569406586695507691, 2.88284002834170367663423348140, 3.51872166651258097621825353372, 3.92466635453073841614723316181, 4.56262939147201569380288848886, 5.38101408225905076987449756363, 5.85997406064744443370246341093, 6.54594806974167425351621693991, 6.81986702064647802816185123443, 7.42877401219022593199367331278, 7.939785486910255471890727563564, 8.444285807094076060195889002243, 9.126851443848936814166943575674, 9.752167845702409155893436600803, 9.869342503231410947193766446648, 10.42232115203152666088650875087, 10.55201879853308613260439031575, 11.60541226325730901937735502807
