| L(s) = 1 | + 4·7-s − 9-s − 4·17-s + 8·23-s − 6·25-s + 12·31-s − 12·41-s − 8·47-s − 2·49-s − 4·63-s + 24·71-s + 20·73-s − 20·79-s + 81-s + 28·89-s + 20·97-s − 20·103-s + 4·113-s − 16·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1/3·9-s − 0.970·17-s + 1.66·23-s − 6/5·25-s + 2.15·31-s − 1.87·41-s − 1.16·47-s − 2/7·49-s − 0.503·63-s + 2.84·71-s + 2.34·73-s − 2.25·79-s + 1/9·81-s + 2.96·89-s + 2.03·97-s − 1.97·103-s + 0.376·113-s − 1.46·119-s + 6/11·121-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 + 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.201958414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.201958414\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.70206185831713788930245676267, −9.916600997518144663362488225329, −9.896673563496253942721087298003, −9.135333050200086696728605841255, −8.821521414159894913227301420852, −8.278326865716987156084613925871, −7.976423808355417881524759290288, −7.83814253456007657478137667272, −6.89746921824129312899974225309, −6.66507801682956302120994236992, −6.30042210312539327070286376552, −5.46098441734282868973399766682, −5.06817891142303539060681617542, −4.75363724414827720741509638280, −4.32986925465745354573808228271, −3.50353875069208416710026970920, −3.03471852115787710647152141334, −2.16416084766877351097921162649, −1.76585453833573443385017399365, −0.77858141589193121347441626816,
0.77858141589193121347441626816, 1.76585453833573443385017399365, 2.16416084766877351097921162649, 3.03471852115787710647152141334, 3.50353875069208416710026970920, 4.32986925465745354573808228271, 4.75363724414827720741509638280, 5.06817891142303539060681617542, 5.46098441734282868973399766682, 6.30042210312539327070286376552, 6.66507801682956302120994236992, 6.89746921824129312899974225309, 7.83814253456007657478137667272, 7.976423808355417881524759290288, 8.278326865716987156084613925871, 8.821521414159894913227301420852, 9.135333050200086696728605841255, 9.896673563496253942721087298003, 9.916600997518144663362488225329, 10.70206185831713788930245676267
