| L(s) = 1 | − 2·3-s + 4-s − 8·7-s + 9-s − 2·12-s − 8·13-s + 16-s − 8·19-s + 16·21-s + 25-s + 4·27-s − 8·28-s + 2·31-s + 36-s + 16·37-s + 16·39-s − 20·43-s − 2·48-s + 34·49-s − 8·52-s + 16·57-s + 28·61-s − 8·63-s + 64-s + 16·67-s − 8·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 3.02·7-s + 1/3·9-s − 0.577·12-s − 2.21·13-s + 1/4·16-s − 1.83·19-s + 3.49·21-s + 1/5·25-s + 0.769·27-s − 1.51·28-s + 0.359·31-s + 1/6·36-s + 2.63·37-s + 2.56·39-s − 3.04·43-s − 0.288·48-s + 34/7·49-s − 1.10·52-s + 2.11·57-s + 3.58·61-s − 1.00·63-s + 1/8·64-s + 1.95·67-s − 0.936·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.140939386265747873328067569806, −7.13014434963814596481540225137, −6.77097595153689876646605531178, −6.72583461739189111327324544278, −6.33719049508577830282130975872, −5.93653080258964708892650187466, −5.21277269593248482625544714347, −5.06637053921005043111949188088, −4.11633325616548011845287467908, −3.86181451968313816241216267886, −2.87299403465911065799617206252, −2.76898105470486532245117845774, −2.16790461724177153358802712730, −0.60676761346407823140058390188, 0,
0.60676761346407823140058390188, 2.16790461724177153358802712730, 2.76898105470486532245117845774, 2.87299403465911065799617206252, 3.86181451968313816241216267886, 4.11633325616548011845287467908, 5.06637053921005043111949188088, 5.21277269593248482625544714347, 5.93653080258964708892650187466, 6.33719049508577830282130975872, 6.72583461739189111327324544278, 6.77097595153689876646605531178, 7.13014434963814596481540225137, 8.140939386265747873328067569806
