| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 5·9-s + 2·10-s + 5·11-s + 4·13-s + 16-s − 5·17-s + 5·18-s + 2·20-s + 5·22-s − 25-s + 4·26-s + 32-s − 5·34-s + 5·36-s − 16·37-s + 2·40-s + 5·44-s + 10·45-s + 5·49-s − 50-s + 4·52-s + 10·55-s + 10·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 5/3·9-s + 0.632·10-s + 1.50·11-s + 1.10·13-s + 1/4·16-s − 1.21·17-s + 1.17·18-s + 0.447·20-s + 1.06·22-s − 1/5·25-s + 0.784·26-s + 0.176·32-s − 0.857·34-s + 5/6·36-s − 2.63·37-s + 0.316·40-s + 0.753·44-s + 1.49·45-s + 5/7·49-s − 0.141·50-s + 0.554·52-s + 1.34·55-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.927354287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.927354287\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 105 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 127 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.693106897828908108247753364178, −7.952030872113872173367866691847, −7.24400594316398621315503187668, −7.01866428971125593812595099427, −6.58741997017797074790016737973, −6.22964068496149410178266735509, −5.78742587874235716738152627165, −5.16293979433082427180753488762, −4.60177403178348512577473878035, −4.10660632241969598906900634621, −3.79277034805909303170184144047, −3.22426006217808700760784232060, −2.14045533678656230614910911624, −1.74171000561321275859235095117, −1.20068436925435355317228040963,
1.20068436925435355317228040963, 1.74171000561321275859235095117, 2.14045533678656230614910911624, 3.22426006217808700760784232060, 3.79277034805909303170184144047, 4.10660632241969598906900634621, 4.60177403178348512577473878035, 5.16293979433082427180753488762, 5.78742587874235716738152627165, 6.22964068496149410178266735509, 6.58741997017797074790016737973, 7.01866428971125593812595099427, 7.24400594316398621315503187668, 7.952030872113872173367866691847, 8.693106897828908108247753364178
