| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s + 12-s + 6·13-s + 15-s − 16-s − 18-s + 20-s − 3·24-s + 25-s − 6·26-s − 27-s − 30-s + 16·31-s − 5·32-s − 36-s − 6·37-s − 6·39-s − 3·40-s + 6·41-s − 45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s + 0.258·15-s − 1/4·16-s − 0.235·18-s + 0.223·20-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.182·30-s + 2.87·31-s − 0.883·32-s − 1/6·36-s − 0.986·37-s − 0.960·39-s − 0.474·40-s + 0.937·41-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8127848591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8127848591\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
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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
−8.905977212366117482335025310366, −8.566266728237752344812164323656, −8.164784813489564550585907462237, −7.85923543569937005011130177851, −7.14084006027180754987274182931, −6.68518872303551465466396243472, −6.16695895670544286351716080604, −5.71340205752869075700678895457, −4.93302791533296556425176498902, −4.60404315210379879095503014039, −3.91910375636700705199954650109, −3.52181352865394515288844372408, −2.54400172959357573602668819491, −1.39941922193662355717119149365, −0.75115206295181260150647173214,
0.75115206295181260150647173214, 1.39941922193662355717119149365, 2.54400172959357573602668819491, 3.52181352865394515288844372408, 3.91910375636700705199954650109, 4.60404315210379879095503014039, 4.93302791533296556425176498902, 5.71340205752869075700678895457, 6.16695895670544286351716080604, 6.68518872303551465466396243472, 7.14084006027180754987274182931, 7.85923543569937005011130177851, 8.164784813489564550585907462237, 8.566266728237752344812164323656, 8.905977212366117482335025310366
