| L(s) = 1 | − 3·9-s − 2·11-s + 2·19-s − 16·29-s + 20·31-s − 6·41-s + 10·49-s + 16·59-s − 20·61-s − 24·71-s − 12·79-s + 18·89-s + 6·99-s − 12·101-s + 28·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 6·171-s + ⋯ |
| L(s) = 1 | − 9-s − 0.603·11-s + 0.458·19-s − 2.97·29-s + 3.59·31-s − 0.937·41-s + 10/7·49-s + 2.08·59-s − 2.56·61-s − 2.84·71-s − 1.35·79-s + 1.90·89-s + 0.603·99-s − 1.19·101-s + 2.68·109-s − 1.72·121-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 − 0.458·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.279381415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.279381415\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.558458105415433116124152083751, −9.172233008589101017225964234474, −8.746598502943036555869545077196, −8.513325630923809575768664163890, −7.894062691419872661851029667603, −7.81236172682069523212773415653, −7.09745507008846527341507454473, −7.01624557358276738604468088384, −6.20563951633160571426882134350, −5.82959345824437763961307185915, −5.70906291585543162607006596579, −5.12463359720506145016660818974, −4.54389805745358666724377655179, −4.30551214951101528837659676053, −3.50108155382465016065228418309, −3.12842516438000579818624841082, −2.66411158964718331863472444671, −2.12685928326681266783326889312, −1.36799207208325963679370961364, −0.44636699284835547915522807559,
0.44636699284835547915522807559, 1.36799207208325963679370961364, 2.12685928326681266783326889312, 2.66411158964718331863472444671, 3.12842516438000579818624841082, 3.50108155382465016065228418309, 4.30551214951101528837659676053, 4.54389805745358666724377655179, 5.12463359720506145016660818974, 5.70906291585543162607006596579, 5.82959345824437763961307185915, 6.20563951633160571426882134350, 7.01624557358276738604468088384, 7.09745507008846527341507454473, 7.81236172682069523212773415653, 7.894062691419872661851029667603, 8.513325630923809575768664163890, 8.746598502943036555869545077196, 9.172233008589101017225964234474, 9.558458105415433116124152083751
