| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 4·8-s + 4·10-s − 2·11-s + 4·13-s + 5·16-s + 8·19-s + 6·20-s − 4·22-s + 12·23-s − 25-s + 8·26-s − 6·31-s + 6·32-s + 16·38-s + 8·40-s − 16·41-s − 6·43-s − 6·44-s + 24·46-s − 49-s − 2·50-s + 12·52-s + 8·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s − 0.603·11-s + 1.10·13-s + 5/4·16-s + 1.83·19-s + 1.34·20-s − 0.852·22-s + 2.50·23-s − 1/5·25-s + 1.56·26-s − 1.07·31-s + 1.06·32-s + 2.59·38-s + 1.26·40-s − 2.49·41-s − 0.914·43-s − 0.904·44-s + 3.53·46-s − 1/7·49-s − 0.282·50-s + 1.66·52-s + 1.09·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.150200886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.150200886\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 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
−10.33673555336647916409547600759, −9.911621336072682294134132614414, −9.688348653705502924542274038665, −9.026992148302749041150542565062, −8.629982367421760075302653426925, −8.206613815292702778824776304582, −7.56121883688658026033254167035, −7.13677148455102846059174443682, −6.72162468443804120859377195207, −6.50553628696783937154504746653, −5.63360031776902051816896841059, −5.44839995654515029882329108714, −5.16769569581449658669563672033, −4.80341290989672380573450592443, −3.73939928058332717238050274978, −3.64088581244328120339801936913, −2.98162866948676686725382735700, −2.53658123581918093172088464834, −1.66225864436190491405609761499, −1.15234147567941063293886720178,
1.15234147567941063293886720178, 1.66225864436190491405609761499, 2.53658123581918093172088464834, 2.98162866948676686725382735700, 3.64088581244328120339801936913, 3.73939928058332717238050274978, 4.80341290989672380573450592443, 5.16769569581449658669563672033, 5.44839995654515029882329108714, 5.63360031776902051816896841059, 6.50553628696783937154504746653, 6.72162468443804120859377195207, 7.13677148455102846059174443682, 7.56121883688658026033254167035, 8.206613815292702778824776304582, 8.629982367421760075302653426925, 9.026992148302749041150542565062, 9.688348653705502924542274038665, 9.911621336072682294134132614414, 10.33673555336647916409547600759
