| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 9-s − 2·11-s + 2·12-s + 4·13-s − 4·16-s + 2·18-s − 4·22-s + 8·23-s − 25-s + 8·26-s + 27-s − 8·32-s − 2·33-s + 2·36-s + 4·39-s − 4·44-s + 16·46-s + 18·47-s − 4·48-s + 11·49-s − 2·50-s + 8·52-s + 2·54-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s − 0.603·11-s + 0.577·12-s + 1.10·13-s − 16-s + 0.471·18-s − 0.852·22-s + 1.66·23-s − 1/5·25-s + 1.56·26-s + 0.192·27-s − 1.41·32-s − 0.348·33-s + 1/3·36-s + 0.640·39-s − 0.603·44-s + 2.35·46-s + 2.62·47-s − 0.577·48-s + 11/7·49-s − 0.282·50-s + 1.10·52-s + 0.272·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.249284223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.249284223\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.028755790922139382896168795846, −8.794614827642349780603070220632, −8.439333560047520544649775927937, −7.66505986486694333159066241147, −7.12629091377271134552327540430, −6.85073886350288365363080648917, −6.12241751609850681792173625996, −5.52282167690075554303727517698, −5.34668146252658900228619106104, −4.53894929342930739368620916750, −3.92764554574875840306496068176, −3.67840796781373175574076394185, −2.61687998381550726785939862479, −2.61182434413499280414159715388, −1.19093389013722883778513109873,
1.19093389013722883778513109873, 2.61182434413499280414159715388, 2.61687998381550726785939862479, 3.67840796781373175574076394185, 3.92764554574875840306496068176, 4.53894929342930739368620916750, 5.34668146252658900228619106104, 5.52282167690075554303727517698, 6.12241751609850681792173625996, 6.85073886350288365363080648917, 7.12629091377271134552327540430, 7.66505986486694333159066241147, 8.439333560047520544649775927937, 8.794614827642349780603070220632, 9.028755790922139382896168795846
