| L(s) = 1 | − 9-s + 8·11-s − 8·19-s + 4·29-s + 20·41-s − 49-s − 24·59-s − 4·61-s + 16·79-s + 81-s + 12·89-s − 8·99-s + 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 8·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.41·11-s − 1.83·19-s + 0.742·29-s + 3.12·41-s − 1/7·49-s − 3.12·59-s − 0.512·61-s + 1.80·79-s + 1/9·81-s + 1.27·89-s − 0.804·99-s + 1.19·101-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.611·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.025934515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.025934515\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.569502510752792407163498201791, −8.345234650155600544744639392132, −7.72217890735056417695975295102, −7.67012416743575133417543176674, −7.04425916576397979110018382153, −6.60143224406263556687461567788, −6.38510867418754062123470359212, −6.10929717588048830989570272199, −5.90081539002089431751090038132, −5.23221814327828545945750829778, −4.55789214010949071059353364390, −4.52322333148571539479265960252, −3.92524624883703762153156211130, −3.89039766349354524633925976209, −3.04189980421983510536309647699, −2.85451970697427067442407974212, −2.02901374417529753535421268966, −1.79837470242905791034530438493, −1.08744664652679649613104132429, −0.55389099522949076589009837758,
0.55389099522949076589009837758, 1.08744664652679649613104132429, 1.79837470242905791034530438493, 2.02901374417529753535421268966, 2.85451970697427067442407974212, 3.04189980421983510536309647699, 3.89039766349354524633925976209, 3.92524624883703762153156211130, 4.52322333148571539479265960252, 4.55789214010949071059353364390, 5.23221814327828545945750829778, 5.90081539002089431751090038132, 6.10929717588048830989570272199, 6.38510867418754062123470359212, 6.60143224406263556687461567788, 7.04425916576397979110018382153, 7.67012416743575133417543176674, 7.72217890735056417695975295102, 8.345234650155600544744639392132, 8.569502510752792407163498201791
