| L(s) = 1 | − 3-s − 3·7-s − 11-s + 5·13-s − 17-s − 5·19-s + 3·21-s − 7·23-s − 4·25-s − 2·27-s + 6·29-s − 4·31-s + 33-s + 37-s − 5·39-s − 6·41-s + 7·43-s − 9·47-s − 6·49-s + 51-s − 3·53-s + 5·57-s + 7·59-s + 7·61-s + 67-s + 7·69-s − 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 0.301·11-s + 1.38·13-s − 0.242·17-s − 1.14·19-s + 0.654·21-s − 1.45·23-s − 4/5·25-s − 0.384·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.164·37-s − 0.800·39-s − 0.937·41-s + 1.06·43-s − 1.31·47-s − 6/7·49-s + 0.140·51-s − 0.412·53-s + 0.662·57-s + 0.911·59-s + 0.896·61-s + 0.122·67-s + 0.842·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 29 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 17 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 66 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 69 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 4 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 98 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T - 35 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T - 77 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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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
−15.2299679204, −14.5685199423, −14.1954386707, −13.5142121056, −13.2270677559, −12.9000443620, −12.3749114063, −11.7979260063, −11.4539545882, −10.8576070567, −10.4844105209, −9.93263033418, −9.56952551759, −8.90972742769, −8.23337753290, −8.08822343149, −7.13047735623, −6.44630577184, −6.26367760730, −5.75747334873, −5.01750949386, −4.06816544367, −3.74146619065, −2.80712170696, −1.75984709209, 0,
1.75984709209, 2.80712170696, 3.74146619065, 4.06816544367, 5.01750949386, 5.75747334873, 6.26367760730, 6.44630577184, 7.13047735623, 8.08822343149, 8.23337753290, 8.90972742769, 9.56952551759, 9.93263033418, 10.4844105209, 10.8576070567, 11.4539545882, 11.7979260063, 12.3749114063, 12.9000443620, 13.2270677559, 13.5142121056, 14.1954386707, 14.5685199423, 15.2299679204
