| L(s) = 1 | − 4-s − 9-s + 16-s − 8·19-s + 8·29-s − 12·31-s + 36-s − 20·41-s + 10·49-s − 24·59-s − 8·61-s − 64-s − 12·71-s + 8·76-s − 20·79-s + 81-s + 4·89-s + 28·101-s − 32·109-s − 8·116-s − 22·121-s + 12·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1/4·16-s − 1.83·19-s + 1.48·29-s − 2.15·31-s + 1/6·36-s − 3.12·41-s + 10/7·49-s − 3.12·59-s − 1.02·61-s − 1/8·64-s − 1.42·71-s + 0.917·76-s − 2.25·79-s + 1/9·81-s + 0.423·89-s + 2.78·101-s − 3.06·109-s − 0.742·116-s − 2·121-s + 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−8.825097305600924684882846311536, −8.495745477700653924604162293949, −7.83130204114576913522229840721, −7.78367525758314996864358333183, −7.11323696493776343337880421612, −6.84691333396427191508576090187, −6.19858416419371220177785976867, −6.17815197663111214311890216504, −5.61740178378633776751199086807, −4.96176379420631685054632513019, −4.88585937677311203950539035886, −4.37309391978305386506785776826, −3.70998212691240318264645158834, −3.65469738875855982980400887343, −2.78147945032466614126753530538, −2.59039265610919749591723163296, −1.56278854575909790660249147422, −1.53706271652864137809279495870, 0, 0,
1.53706271652864137809279495870, 1.56278854575909790660249147422, 2.59039265610919749591723163296, 2.78147945032466614126753530538, 3.65469738875855982980400887343, 3.70998212691240318264645158834, 4.37309391978305386506785776826, 4.88585937677311203950539035886, 4.96176379420631685054632513019, 5.61740178378633776751199086807, 6.17815197663111214311890216504, 6.19858416419371220177785976867, 6.84691333396427191508576090187, 7.11323696493776343337880421612, 7.78367525758314996864358333183, 7.83130204114576913522229840721, 8.495745477700653924604162293949, 8.825097305600924684882846311536
