| L(s) = 1 | + 4·5-s + 4·17-s + 2·19-s + 2·25-s − 12·31-s + 6·49-s − 16·59-s − 16·61-s + 4·67-s − 16·71-s + 16·79-s + 16·85-s + 8·95-s − 12·101-s + 16·103-s + 40·107-s + 20·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.970·17-s + 0.458·19-s + 2/5·25-s − 2.15·31-s + 6/7·49-s − 2.08·59-s − 2.04·61-s + 0.488·67-s − 1.89·71-s + 1.80·79-s + 1.73·85-s + 0.820·95-s − 1.19·101-s + 1.57·103-s + 3.86·107-s + 1.81·121-s − 2.50·125-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 − 3.85·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.340814497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.340814497\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 186 T^{2} + p^{2} T^{4} \) |
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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
−9.254953751879335105640281244228, −8.803836303869388025887237990287, −8.314135296785240751337418251325, −7.61204689926753097539005080221, −7.59024009090100770350262745389, −7.28228606117241987537414687676, −6.56653721126262848569220390314, −6.21323297088349995759190368928, −5.91695232788834883211118404340, −5.57221263482427776957652132679, −5.36368366270603245986997134831, −4.77119336982004184635187421274, −4.38894172675062966291398448840, −3.70746400731045660772215764563, −3.29888196616525037243434017917, −2.90984757051743393826411315754, −2.21803184637689717829514482767, −1.69550106553071754436707702664, −1.58969387208120941012034376764, −0.56960718636903515176045558222,
0.56960718636903515176045558222, 1.58969387208120941012034376764, 1.69550106553071754436707702664, 2.21803184637689717829514482767, 2.90984757051743393826411315754, 3.29888196616525037243434017917, 3.70746400731045660772215764563, 4.38894172675062966291398448840, 4.77119336982004184635187421274, 5.36368366270603245986997134831, 5.57221263482427776957652132679, 5.91695232788834883211118404340, 6.21323297088349995759190368928, 6.56653721126262848569220390314, 7.28228606117241987537414687676, 7.59024009090100770350262745389, 7.61204689926753097539005080221, 8.314135296785240751337418251325, 8.803836303869388025887237990287, 9.254953751879335105640281244228
