| L(s) = 1 | + 2·5-s + 4·11-s − 12·19-s − 25-s − 12·29-s − 20·31-s − 12·41-s − 49-s + 8·55-s − 12·61-s + 12·71-s + 8·79-s − 12·89-s − 24·95-s + 20·101-s − 20·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s − 40·155-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s − 2.75·19-s − 1/5·25-s − 2.22·29-s − 3.59·31-s − 1.87·41-s − 1/7·49-s + 1.07·55-s − 1.53·61-s + 1.42·71-s + 0.900·79-s − 1.27·89-s − 2.46·95-s + 1.99·101-s − 1.91·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s − 3.21·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−7.974592844977877167482714102690, −7.82599250694865143822177648553, −7.12521482891508758739054378357, −6.92016920677050672860477405701, −6.75598933456243583547108428708, −6.13066635692707908050847077647, −5.82608940163686755989898972603, −5.77713655929081038602991507530, −5.09398190124377666658123984641, −4.86242002164603001512925158734, −4.20590595823986784386355259747, −3.85975039591538067771217462743, −3.64002613222366141954509148812, −3.26534209085298745783866657602, −2.25660487287582755218612286140, −2.12803521529952373093342773571, −1.72916586633351143145627446274, −1.39983149021245901505943006936, 0, 0,
1.39983149021245901505943006936, 1.72916586633351143145627446274, 2.12803521529952373093342773571, 2.25660487287582755218612286140, 3.26534209085298745783866657602, 3.64002613222366141954509148812, 3.85975039591538067771217462743, 4.20590595823986784386355259747, 4.86242002164603001512925158734, 5.09398190124377666658123984641, 5.77713655929081038602991507530, 5.82608940163686755989898972603, 6.13066635692707908050847077647, 6.75598933456243583547108428708, 6.92016920677050672860477405701, 7.12521482891508758739054378357, 7.82599250694865143822177648553, 7.974592844977877167482714102690
