| L(s) = 1 | + 3·2-s + 3-s + 4·4-s + 3·6-s − 3·7-s + 3·8-s + 3·9-s + 6·11-s + 4·12-s − 2·13-s − 9·14-s + 3·16-s + 17-s + 9·18-s − 12·19-s − 3·21-s + 18·22-s − 3·23-s + 3·24-s − 2·25-s − 6·26-s + 8·27-s − 12·28-s + 6·29-s + 6·32-s + 6·33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 2·4-s + 1.22·6-s − 1.13·7-s + 1.06·8-s + 9-s + 1.80·11-s + 1.15·12-s − 0.554·13-s − 2.40·14-s + 3/4·16-s + 0.242·17-s + 2.12·18-s − 2.75·19-s − 0.654·21-s + 3.83·22-s − 0.625·23-s + 0.612·24-s − 2/5·25-s − 1.17·26-s + 1.53·27-s − 2.26·28-s + 1.11·29-s + 1.06·32-s + 1.04·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.401272094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.401272094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 24 T + 251 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 27 T + 332 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−12.42428001476611770867950146508, −12.28822971065770917936464457094, −12.17538969467332375127440536641, −11.30521884889324131841692181805, −10.62375402997325204265505532754, −10.05712592776468210883289924204, −9.781963712146714981725774732044, −9.157693178646631446199810357649, −8.443879735045716896433627850807, −8.158522928650359448948914865186, −6.90075954042836357093825182971, −6.89093344146662211295774366512, −6.15021531138913737645069045726, −5.99219888420951799161880915508, −4.74405182140688908786367777604, −4.42853293033965041785306056281, −4.06280221744515905330578516630, −3.43950741555442684516321834282, −2.78333645780879759283830866153, −1.73299281478578995624859216809,
1.73299281478578995624859216809, 2.78333645780879759283830866153, 3.43950741555442684516321834282, 4.06280221744515905330578516630, 4.42853293033965041785306056281, 4.74405182140688908786367777604, 5.99219888420951799161880915508, 6.15021531138913737645069045726, 6.89093344146662211295774366512, 6.90075954042836357093825182971, 8.158522928650359448948914865186, 8.443879735045716896433627850807, 9.157693178646631446199810357649, 9.781963712146714981725774732044, 10.05712592776468210883289924204, 10.62375402997325204265505532754, 11.30521884889324131841692181805, 12.17538969467332375127440536641, 12.28822971065770917936464457094, 12.42428001476611770867950146508
