L(s) = 1 | + 16·17-s − 6·25-s + 16·41-s − 14·49-s − 12·73-s − 32·89-s − 36·97-s + 32·113-s + 22·121-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 + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 3.88·17-s − 6/5·25-s + 2.49·41-s − 2·49-s − 1.40·73-s − 3.39·89-s − 3.65·97-s + 3.01·113-s + 2·121-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 − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.646260255\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.646260255\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−9.548848031972466972528047632167, −8.563614314939728499388923405126, −8.466315049049263157483367972277, −8.008665384019597065619698123983, −7.56723502154780010094923636286, −7.44561399182354586601083013460, −7.09599337490182052157137137928, −6.27750501971151210359225889756, −6.02723024285242528483102462633, −5.57748949513655342230201835766, −5.49391613207571551538128644678, −4.88167376255002505883363300900, −4.28999318051224301124193831558, −3.89507888397287736947491483187, −3.45193035642142507051958864370, −2.91010175768751731444170735721, −2.70787924317443891395253245656, −1.60124216691692040003103208613, −1.39901299164329195055601839800, −0.59706028858537316754310186535,
0.59706028858537316754310186535, 1.39901299164329195055601839800, 1.60124216691692040003103208613, 2.70787924317443891395253245656, 2.91010175768751731444170735721, 3.45193035642142507051958864370, 3.89507888397287736947491483187, 4.28999318051224301124193831558, 4.88167376255002505883363300900, 5.49391613207571551538128644678, 5.57748949513655342230201835766, 6.02723024285242528483102462633, 6.27750501971151210359225889756, 7.09599337490182052157137137928, 7.44561399182354586601083013460, 7.56723502154780010094923636286, 8.008665384019597065619698123983, 8.466315049049263157483367972277, 8.563614314939728499388923405126, 9.548848031972466972528047632167