| L(s) = 1 | − 3-s + 4-s − 2·7-s − 2·9-s − 12-s − 6·13-s + 16-s − 2·19-s + 2·21-s + 25-s + 5·27-s − 2·28-s + 20·31-s − 2·36-s + 4·37-s + 6·39-s + 12·43-s − 48-s − 11·49-s − 6·52-s + 2·57-s − 8·61-s + 4·63-s + 64-s + 14·67-s − 18·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.755·7-s − 2/3·9-s − 0.288·12-s − 1.66·13-s + 1/4·16-s − 0.458·19-s + 0.436·21-s + 1/5·25-s + 0.962·27-s − 0.377·28-s + 3.59·31-s − 1/3·36-s + 0.657·37-s + 0.960·39-s + 1.82·43-s − 0.144·48-s − 1.57·49-s − 0.832·52-s + 0.264·57-s − 1.02·61-s + 0.503·63-s + 1/8·64-s + 1.71·67-s − 2.10·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−8.442256105932250546147711653462, −8.079561289589277419191527740912, −7.62707102022517352540213975378, −6.88551821385603277867593120567, −6.77368415127665659188399947657, −6.05512553818884273984844633294, −5.93621501809148728100755282492, −5.19515652654869985902764930365, −4.49281115162991661669735579088, −4.42087197415110762692909845352, −3.23779582731709228828793616424, −2.60972442256408021133851097518, −2.58032194053261503171562760419, −1.14942253786144881590474930387, 0,
1.14942253786144881590474930387, 2.58032194053261503171562760419, 2.60972442256408021133851097518, 3.23779582731709228828793616424, 4.42087197415110762692909845352, 4.49281115162991661669735579088, 5.19515652654869985902764930365, 5.93621501809148728100755282492, 6.05512553818884273984844633294, 6.77368415127665659188399947657, 6.88551821385603277867593120567, 7.62707102022517352540213975378, 8.079561289589277419191527740912, 8.442256105932250546147711653462
