| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s + 5-s − 4·6-s − 3·9-s + 2·10-s − 11-s − 4·12-s − 8·13-s − 2·15-s − 4·16-s + 4·17-s − 6·18-s + 2·20-s − 2·22-s − 2·23-s − 4·25-s − 16·26-s + 14·27-s − 4·30-s − 8·32-s + 2·33-s + 8·34-s − 6·36-s + 16·39-s − 2·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s + 0.447·5-s − 1.63·6-s − 9-s + 0.632·10-s − 0.301·11-s − 1.15·12-s − 2.21·13-s − 0.516·15-s − 16-s + 0.970·17-s − 1.41·18-s + 0.447·20-s − 0.426·22-s − 0.417·23-s − 4/5·25-s − 3.13·26-s + 2.69·27-s − 0.730·30-s − 1.41·32-s + 0.348·33-s + 1.37·34-s − 36-s + 2.56·39-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 532400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147836550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147836550\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.394238320905654147490500873996, −7.78789607043645765017922373874, −7.49534540311409429819739808064, −6.89747784534263149853829967422, −6.23870064789214498217427813253, −6.10845204313762133714030223921, −5.41175071356617719099292784896, −5.40055166000655653433368694834, −4.88140290332121063087048287423, −4.44574490854809981150981932393, −3.68709007429289538959412107978, −2.96905370396904801865789065276, −2.63044898935838963010520851422, −2.00337434233232539103171654262, −0.44102589544325217691153494662,
0.44102589544325217691153494662, 2.00337434233232539103171654262, 2.63044898935838963010520851422, 2.96905370396904801865789065276, 3.68709007429289538959412107978, 4.44574490854809981150981932393, 4.88140290332121063087048287423, 5.40055166000655653433368694834, 5.41175071356617719099292784896, 6.10845204313762133714030223921, 6.23870064789214498217427813253, 6.89747784534263149853829967422, 7.49534540311409429819739808064, 7.78789607043645765017922373874, 8.394238320905654147490500873996
