| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s + 6·6-s + 4·9-s − 6·11-s − 6·12-s − 4·16-s − 7·17-s − 8·18-s − 4·19-s + 12·22-s − 2·25-s + 8·32-s + 18·33-s + 14·34-s + 8·36-s + 8·38-s − 4·41-s + 8·43-s − 12·44-s + 12·48-s − 2·49-s + 4·50-s + 21·51-s + 12·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s + 4/3·9-s − 1.80·11-s − 1.73·12-s − 16-s − 1.69·17-s − 1.88·18-s − 0.917·19-s + 2.55·22-s − 2/5·25-s + 1.41·32-s + 3.13·33-s + 2.40·34-s + 4/3·36-s + 1.29·38-s − 0.624·41-s + 1.21·43-s − 1.80·44-s + 1.73·48-s − 2/7·49-s + 0.565·50-s + 2.94·51-s + 1.58·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−12.21042573364748676074371926432, −11.51872936506842932746880271040, −10.98133269166747767917912100925, −10.61403928556650556894125675254, −10.33963011208683484979467530412, −9.438558187505883569273911834359, −8.733721189989508905862760810801, −8.082402023422120616644237332671, −7.38360144889446860906003907987, −6.62198346346856840746921529147, −6.01678572040208163918026990344, −5.08860375066156614816012066413, −4.46631018668298863468822429117, −2.33446241377498533200596611163, 0,
2.33446241377498533200596611163, 4.46631018668298863468822429117, 5.08860375066156614816012066413, 6.01678572040208163918026990344, 6.62198346346856840746921529147, 7.38360144889446860906003907987, 8.082402023422120616644237332671, 8.733721189989508905862760810801, 9.438558187505883569273911834359, 10.33963011208683484979467530412, 10.61403928556650556894125675254, 10.98133269166747767917912100925, 11.51872936506842932746880271040, 12.21042573364748676074371926432
