| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 6·11-s + 15-s − 2·21-s + 25-s − 27-s + 6·33-s − 2·35-s + 8·43-s − 45-s − 2·49-s + 12·53-s + 6·55-s − 18·59-s + 4·61-s + 2·63-s − 4·67-s − 12·71-s − 75-s − 12·77-s + 81-s − 6·99-s − 10·103-s + 2·105-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.258·15-s − 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.04·33-s − 0.338·35-s + 1.21·43-s − 0.149·45-s − 2/7·49-s + 1.64·53-s + 0.809·55-s − 2.34·59-s + 0.512·61-s + 0.251·63-s − 0.488·67-s − 1.42·71-s − 0.115·75-s − 1.36·77-s + 1/9·81-s − 0.603·99-s − 0.985·103-s + 0.195·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.683893751284429556749676941632, −8.242498904373773901863115668516, −7.83298189622501308320383508503, −7.38361830089163477437182996650, −7.11172911542884038573278201249, −6.25930713551989881226908856716, −5.78271512389619020996117690528, −5.34037661038038998021641847667, −4.79150066353033527558648259761, −4.43669898042292026640195142146, −3.70528607298888800971732556574, −2.86820919626464618858559542612, −2.32769520098898456110671665937, −1.29044932946073407894236523594, 0,
1.29044932946073407894236523594, 2.32769520098898456110671665937, 2.86820919626464618858559542612, 3.70528607298888800971732556574, 4.43669898042292026640195142146, 4.79150066353033527558648259761, 5.34037661038038998021641847667, 5.78271512389619020996117690528, 6.25930713551989881226908856716, 7.11172911542884038573278201249, 7.38361830089163477437182996650, 7.83298189622501308320383508503, 8.242498904373773901863115668516, 8.683893751284429556749676941632
