| L(s) = 1 | − 3-s − 2·5-s − 2·9-s − 2·11-s + 8·13-s + 2·15-s − 4·16-s + 4·17-s + 23-s − 7·25-s + 5·27-s + 14·31-s + 2·33-s − 8·39-s + 4·45-s + 4·48-s − 10·49-s − 4·51-s + 12·53-s + 4·55-s − 16·65-s − 69-s + 8·73-s + 7·75-s + 8·80-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 2/3·9-s − 0.603·11-s + 2.21·13-s + 0.516·15-s − 16-s + 0.970·17-s + 0.208·23-s − 7/5·25-s + 0.962·27-s + 2.51·31-s + 0.348·33-s − 1.28·39-s + 0.596·45-s + 0.577·48-s − 1.42·49-s − 0.560·51-s + 1.64·53-s + 0.539·55-s − 1.98·65-s − 0.120·69-s + 0.936·73-s + 0.808·75-s + 0.894·80-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.024794649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024794649\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.228854358433494441874279371750, −8.212973485084645328991374926218, −7.72626049705676692250966013712, −6.87252557874530003114725570070, −6.66503987304844816686807720890, −6.10199205171023259542319673445, −5.66407660024727617127120629681, −5.35806526558242506412465562277, −4.54560153511675547350568265865, −4.19085115174357929506261869806, −3.61425154305823041342611871942, −3.09000916592887094247460325417, −2.48240948578189648627819728306, −1.42318267584838641544176432302, −0.57357212901825478877549470562,
0.57357212901825478877549470562, 1.42318267584838641544176432302, 2.48240948578189648627819728306, 3.09000916592887094247460325417, 3.61425154305823041342611871942, 4.19085115174357929506261869806, 4.54560153511675547350568265865, 5.35806526558242506412465562277, 5.66407660024727617127120629681, 6.10199205171023259542319673445, 6.66503987304844816686807720890, 6.87252557874530003114725570070, 7.72626049705676692250966013712, 8.212973485084645328991374926218, 8.228854358433494441874279371750
