L(s) = 1 | − 6·17-s − 6·25-s + 4·29-s − 10·37-s + 18·41-s + 14·49-s + 8·53-s + 22·61-s − 10·73-s − 9·81-s + 26·89-s − 26·97-s + 18·101-s + 30·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 1.45·17-s − 6/5·25-s + 0.742·29-s − 1.64·37-s + 2.81·41-s + 2·49-s + 1.09·53-s + 2.81·61-s − 1.17·73-s − 81-s + 2.75·89-s − 2.63·97-s + 1.79·101-s + 2.82·113-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 − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.481340666\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.481340666\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.24563196122341242475579558411, −10.72674065320852401537343863362, −10.52042302064175550405915224930, −9.869440332118278822742767131161, −9.565844329015149439834428752666, −8.816357121070883728515325784981, −8.732491796553252599368702627843, −8.230792534520394323590633299873, −7.37878097640437458870999633399, −7.29883027474054552875378692434, −6.68619918827544669799741195667, −6.06713722103888559426951018831, −5.68499667047805118350883385239, −5.12515621359559477498284511495, −4.25426998564930941154292540488, −4.17338544663939196008426562704, −3.35078920772161511875118521174, −2.41583172862249535361261449715, −2.09398799371257205659214757961, −0.75860764462431132409728642856,
0.75860764462431132409728642856, 2.09398799371257205659214757961, 2.41583172862249535361261449715, 3.35078920772161511875118521174, 4.17338544663939196008426562704, 4.25426998564930941154292540488, 5.12515621359559477498284511495, 5.68499667047805118350883385239, 6.06713722103888559426951018831, 6.68619918827544669799741195667, 7.29883027474054552875378692434, 7.37878097640437458870999633399, 8.230792534520394323590633299873, 8.732491796553252599368702627843, 8.816357121070883728515325784981, 9.565844329015149439834428752666, 9.869440332118278822742767131161, 10.52042302064175550405915224930, 10.72674065320852401537343863362, 11.24563196122341242475579558411