L(s) = 1 | − 4·3-s + 4·7-s + 8·9-s + 16·11-s − 6·13-s − 14·17-s − 16·21-s − 4·23-s − 36·27-s − 104·31-s − 64·33-s + 6·37-s + 24·39-s − 16·41-s − 84·43-s − 36·47-s + 8·49-s + 56·51-s − 106·53-s − 96·61-s + 32·63-s + 124·67-s + 16·69-s + 56·71-s + 94·73-s + 64·77-s + 143·81-s + ⋯ |
L(s) = 1 | − 4/3·3-s + 4/7·7-s + 8/9·9-s + 1.45·11-s − 0.461·13-s − 0.823·17-s − 0.761·21-s − 0.173·23-s − 4/3·27-s − 3.35·31-s − 1.93·33-s + 6/37·37-s + 8/13·39-s − 0.390·41-s − 1.95·43-s − 0.765·47-s + 8/49·49-s + 1.09·51-s − 2·53-s − 1.57·61-s + 0.507·63-s + 1.85·67-s + 0.231·69-s + 0.788·71-s + 1.28·73-s + 0.831·77-s + 1.76·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7809207978\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7809207978\) |
\(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 52 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 84 T + 3528 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 124 T + 7688 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 94 T + 4418 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9442 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.36219301202110009582046686494, −11.02925757659418387936956820616, −10.62568598103755953516354118666, −9.762342916850009284525604533544, −9.694841157114394839294440425383, −8.874035414996817492955701019736, −8.858393717482030118473741855376, −7.72371721650496784625870505845, −7.69332304342386810225971950094, −6.81494813700288390885837981447, −6.58802317176632074032570216965, −6.06944491546468071274347614546, −5.50709431715243810081971644691, −4.88041120016805143186330477747, −4.71301936461890810315053742308, −3.63743919069598103873364085530, −3.56679001993819847741093433042, −1.86793173718946404323648141080, −1.79004319827305602227911783764, −0.41548441986364967330235315004,
0.41548441986364967330235315004, 1.79004319827305602227911783764, 1.86793173718946404323648141080, 3.56679001993819847741093433042, 3.63743919069598103873364085530, 4.71301936461890810315053742308, 4.88041120016805143186330477747, 5.50709431715243810081971644691, 6.06944491546468071274347614546, 6.58802317176632074032570216965, 6.81494813700288390885837981447, 7.69332304342386810225971950094, 7.72371721650496784625870505845, 8.858393717482030118473741855376, 8.874035414996817492955701019736, 9.694841157114394839294440425383, 9.762342916850009284525604533544, 10.62568598103755953516354118666, 11.02925757659418387936956820616, 11.36219301202110009582046686494