| L(s) = 1 | − 4·13-s + 8·19-s + 5·25-s − 4·31-s + 10·37-s + 16·43-s + 14·61-s + 16·67-s − 10·73-s + 4·79-s − 28·97-s + 20·103-s − 2·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.10·13-s + 1.83·19-s + 25-s − 0.718·31-s + 1.64·37-s + 2.43·43-s + 1.79·61-s + 1.95·67-s − 1.17·73-s + 0.450·79-s − 2.84·97-s + 1.97·103-s − 0.191·109-s + 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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.527861701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.527861701\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.530044344664672823687370209582, −9.361189171956335614796828198990, −8.516798118005930908986577574619, −8.472291674775557307128830219238, −7.86147694135727985556909256989, −7.39745551303209634332365244822, −7.07345698940820125811355153222, −7.07050729325715355674057844797, −6.15921684507740690464974393958, −5.88020959733861392847866171818, −5.36215989589383442813981310633, −5.13500414511092058323493125901, −4.45113063728847987930472033292, −4.26571316277168790413314224553, −3.46801658604687274419207993817, −3.14774433695310256136608703272, −2.45455288832451320950951149337, −2.22379300296538333079826554310, −1.14673234245407380756870781757, −0.69335406972705442880699528951,
0.69335406972705442880699528951, 1.14673234245407380756870781757, 2.22379300296538333079826554310, 2.45455288832451320950951149337, 3.14774433695310256136608703272, 3.46801658604687274419207993817, 4.26571316277168790413314224553, 4.45113063728847987930472033292, 5.13500414511092058323493125901, 5.36215989589383442813981310633, 5.88020959733861392847866171818, 6.15921684507740690464974393958, 7.07050729325715355674057844797, 7.07345698940820125811355153222, 7.39745551303209634332365244822, 7.86147694135727985556909256989, 8.472291674775557307128830219238, 8.516798118005930908986577574619, 9.361189171956335614796828198990, 9.530044344664672823687370209582
