| L(s) = 1 | − 3-s − 2·4-s − 7-s + 9-s + 2·12-s + 5·13-s + 13·19-s + 21-s + 3·25-s − 27-s + 2·28-s + 13·31-s − 2·36-s + 13·37-s − 5·39-s − 14·43-s − 13·49-s − 10·52-s − 13·57-s + 6·61-s − 63-s + 8·64-s − 3·67-s + 9·73-s − 3·75-s − 26·76-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 1.38·13-s + 2.98·19-s + 0.218·21-s + 3/5·25-s − 0.192·27-s + 0.377·28-s + 2.33·31-s − 1/3·36-s + 2.13·37-s − 0.800·39-s − 2.13·43-s − 1.85·49-s − 1.38·52-s − 1.72·57-s + 0.768·61-s − 0.125·63-s + 64-s − 0.366·67-s + 1.05·73-s − 0.346·75-s − 2.98·76-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321273 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321273 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.285092307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.285092307\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) |
| 163 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 20 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.795594384865011387711348328180, −8.299470714853511324662679483640, −7.968319850357172049156039945125, −7.47048910123912809704684103283, −6.70510554839776128308567376364, −6.45096611237985360861565966731, −5.96026370484332304967137462179, −5.28862796534101678057046141469, −4.93206518472978227205141437093, −4.52836834284055157116854017730, −3.75586164318120136732367217753, −3.29440406760256609167356914254, −2.73366493304961585524258222402, −1.32850106519656943939372054661, −0.807214935163006539243251492250,
0.807214935163006539243251492250, 1.32850106519656943939372054661, 2.73366493304961585524258222402, 3.29440406760256609167356914254, 3.75586164318120136732367217753, 4.52836834284055157116854017730, 4.93206518472978227205141437093, 5.28862796534101678057046141469, 5.96026370484332304967137462179, 6.45096611237985360861565966731, 6.70510554839776128308567376364, 7.47048910123912809704684103283, 7.968319850357172049156039945125, 8.299470714853511324662679483640, 8.795594384865011387711348328180
