| L(s) = 1 | + 3-s − 6·7-s + 9-s − 4·13-s − 4·16-s + 12·19-s − 6·21-s − 9·25-s + 27-s + 4·31-s − 14·37-s − 4·39-s − 20·43-s − 4·48-s + 13·49-s + 12·57-s − 20·61-s − 6·63-s + 20·67-s − 20·73-s − 9·75-s + 34·79-s + 81-s + 24·91-s + 4·93-s + 2·97-s + 22·103-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.26·7-s + 1/3·9-s − 1.10·13-s − 16-s + 2.75·19-s − 1.30·21-s − 9/5·25-s + 0.192·27-s + 0.718·31-s − 2.30·37-s − 0.640·39-s − 3.04·43-s − 0.577·48-s + 13/7·49-s + 1.58·57-s − 2.56·61-s − 0.755·63-s + 2.44·67-s − 2.34·73-s − 1.03·75-s + 3.82·79-s + 1/9·81-s + 2.51·91-s + 0.414·93-s + 0.203·97-s + 2.16·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−9.827535245480289108967102730184, −9.149156412188436581192825917436, −9.085897389420589721942660476599, −8.100801554629335905381784580980, −7.52539857035144694302700552711, −7.19598013314039086114174891084, −6.46487870715497229983731599323, −6.33758757673418292336292093348, −5.14885571821443324588162204875, −5.05201755691324140880779008098, −3.77257881491072191084851286374, −3.33358434652105144081845579676, −2.93657054326743540839942565323, −1.89809557201657584670009427079, 0,
1.89809557201657584670009427079, 2.93657054326743540839942565323, 3.33358434652105144081845579676, 3.77257881491072191084851286374, 5.05201755691324140880779008098, 5.14885571821443324588162204875, 6.33758757673418292336292093348, 6.46487870715497229983731599323, 7.19598013314039086114174891084, 7.52539857035144694302700552711, 8.100801554629335905381784580980, 9.085897389420589721942660476599, 9.149156412188436581192825917436, 9.827535245480289108967102730184
