L(s) = 1 | + 3-s + 4-s + 2·7-s + 9-s + 12-s + 4·13-s + 16-s + 4·19-s + 2·21-s − 10·25-s + 27-s + 2·28-s + 4·31-s + 36-s + 4·37-s + 4·39-s − 8·43-s + 48-s + 3·49-s + 4·52-s + 4·57-s + 4·61-s + 2·63-s + 64-s − 8·67-s − 8·73-s − 10·75-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.917·19-s + 0.436·21-s − 2·25-s + 0.192·27-s + 0.377·28-s + 0.718·31-s + 1/6·36-s + 0.657·37-s + 0.640·39-s − 1.21·43-s + 0.144·48-s + 3/7·49-s + 0.554·52-s + 0.529·57-s + 0.512·61-s + 0.251·63-s + 1/8·64-s − 0.977·67-s − 0.936·73-s − 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.451405122\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.451405122\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.158830011188341869727925681893, −8.104024357108211588654438142125, −7.46173924472506058951788867760, −7.27269450626182225527303312385, −6.40050362102582630929450786700, −6.27935085429088375420913933534, −5.57921544774938402027851071868, −5.25821839364078131696035606890, −4.44481069307592434914952580537, −4.14440200645550446493587121840, −3.37789225057720607500704079136, −3.12443402203060754923261311981, −2.16035280583295696954900337452, −1.77631258837245971519374380176, −0.960161767014088955334438808530,
0.960161767014088955334438808530, 1.77631258837245971519374380176, 2.16035280583295696954900337452, 3.12443402203060754923261311981, 3.37789225057720607500704079136, 4.14440200645550446493587121840, 4.44481069307592434914952580537, 5.25821839364078131696035606890, 5.57921544774938402027851071868, 6.27935085429088375420913933534, 6.40050362102582630929450786700, 7.27269450626182225527303312385, 7.46173924472506058951788867760, 8.104024357108211588654438142125, 8.158830011188341869727925681893