| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 3·9-s − 2·14-s + 16-s + 3·18-s − 2·19-s + 6·25-s + 2·28-s − 6·29-s − 32-s − 3·36-s + 2·38-s − 12·41-s − 12·43-s − 2·49-s − 6·50-s + 14·53-s − 2·56-s + 6·58-s − 8·59-s + 16·61-s − 6·63-s + 64-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 9-s − 0.534·14-s + 1/4·16-s + 0.707·18-s − 0.458·19-s + 6/5·25-s + 0.377·28-s − 1.11·29-s − 0.176·32-s − 1/2·36-s + 0.324·38-s − 1.87·41-s − 1.82·43-s − 2/7·49-s − 0.848·50-s + 1.92·53-s − 0.267·56-s + 0.787·58-s − 1.04·59-s + 2.04·61-s − 0.755·63-s + 1/8·64-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.048216588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048216588\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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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
−8.382031367848189614284888899548, −7.946460775026354451010475030496, −7.33980507106402302280971279850, −6.94658827968775692241809795813, −6.62972429418679471198571735198, −5.95381429655996430279682187921, −5.54767665998733378782648856168, −5.11414662503911675414681345387, −4.63916005926183098819435626839, −3.96411106196236159167634968511, −3.24520588568880986669664433480, −2.93526876263737589302358491872, −2.01613011617934039613777940763, −1.67211399362966394964162435763, −0.53978280975700209971582834519,
0.53978280975700209971582834519, 1.67211399362966394964162435763, 2.01613011617934039613777940763, 2.93526876263737589302358491872, 3.24520588568880986669664433480, 3.96411106196236159167634968511, 4.63916005926183098819435626839, 5.11414662503911675414681345387, 5.54767665998733378782648856168, 5.95381429655996430279682187921, 6.62972429418679471198571735198, 6.94658827968775692241809795813, 7.33980507106402302280971279850, 7.946460775026354451010475030496, 8.382031367848189614284888899548
