| L(s) = 1 | + 3-s − 3·4-s + 9-s − 3·12-s + 2·13-s + 5·16-s − 8·19-s − 6·25-s + 27-s − 3·36-s + 12·37-s + 2·39-s + 8·43-s + 5·48-s − 14·49-s − 6·52-s − 8·57-s + 28·61-s − 3·64-s − 24·67-s − 12·73-s − 6·75-s + 24·76-s + 16·79-s + 81-s − 28·97-s + 18·100-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3/2·4-s + 1/3·9-s − 0.866·12-s + 0.554·13-s + 5/4·16-s − 1.83·19-s − 6/5·25-s + 0.192·27-s − 1/2·36-s + 1.97·37-s + 0.320·39-s + 1.21·43-s + 0.721·48-s − 2·49-s − 0.832·52-s − 1.05·57-s + 3.58·61-s − 3/8·64-s − 2.93·67-s − 1.40·73-s − 0.692·75-s + 2.75·76-s + 1.80·79-s + 1/9·81-s − 2.84·97-s + 9/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552123 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552123 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.279842457480330856862908947423, −7.971709728009125280143648464054, −7.55062673006105882301796480772, −6.91274116401925090743914373893, −6.14885963657490762630313444248, −6.09840667273379561083677910559, −5.39216574159615869785069323377, −4.65912153519243395995881297600, −4.44237312587668279705595731913, −3.89641224685561841382047565850, −3.58566955557413367386868416593, −2.66792787949502047211352973601, −2.09639822932402252350030417761, −1.12169940393228089488805712125, 0,
1.12169940393228089488805712125, 2.09639822932402252350030417761, 2.66792787949502047211352973601, 3.58566955557413367386868416593, 3.89641224685561841382047565850, 4.44237312587668279705595731913, 4.65912153519243395995881297600, 5.39216574159615869785069323377, 6.09840667273379561083677910559, 6.14885963657490762630313444248, 6.91274116401925090743914373893, 7.55062673006105882301796480772, 7.971709728009125280143648464054, 8.279842457480330856862908947423
