| L(s) = 1 | − 2·3-s − 3·9-s − 6·11-s + 4·17-s − 4·19-s − 10·25-s + 14·27-s + 12·33-s + 14·41-s + 8·43-s − 5·49-s − 8·51-s + 8·57-s + 16·59-s + 24·67-s − 26·73-s + 20·75-s − 4·81-s − 2·83-s − 4·89-s − 24·97-s + 18·99-s + 24·107-s + 12·113-s + 5·121-s − 28·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s − 1.80·11-s + 0.970·17-s − 0.917·19-s − 2·25-s + 2.69·27-s + 2.08·33-s + 2.18·41-s + 1.21·43-s − 5/7·49-s − 1.12·51-s + 1.05·57-s + 2.08·59-s + 2.93·67-s − 3.04·73-s + 2.30·75-s − 4/9·81-s − 0.219·83-s − 0.423·89-s − 2.43·97-s + 1.80·99-s + 2.32·107-s + 1.12·113-s + 5/11·121-s − 2.52·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−8.147632424012807415917485260643, −7.60181697406565497237849590709, −7.34822351800845502523348597298, −6.56693056444873579906963173743, −6.08449220356234607481996960226, −5.68117327326155315459284776997, −5.54270171583374531541087657986, −5.12156932676137442309894691185, −4.39751259392865511282852863898, −3.93279504709150990560375020684, −3.10384120071878195269767964060, −2.58862548114280755949345221322, −2.14458517210902147128578216964, −0.78387950046628484411267942714, 0,
0.78387950046628484411267942714, 2.14458517210902147128578216964, 2.58862548114280755949345221322, 3.10384120071878195269767964060, 3.93279504709150990560375020684, 4.39751259392865511282852863898, 5.12156932676137442309894691185, 5.54270171583374531541087657986, 5.68117327326155315459284776997, 6.08449220356234607481996960226, 6.56693056444873579906963173743, 7.34822351800845502523348597298, 7.60181697406565497237849590709, 8.147632424012807415917485260643
