| L(s) = 1 | − 4·3-s − 7-s + 6·9-s + 4·19-s + 4·21-s + 6·25-s + 4·27-s + 4·29-s − 8·31-s − 12·37-s + 8·47-s + 49-s − 20·53-s − 16·57-s − 12·59-s − 6·63-s − 24·75-s − 37·81-s − 12·83-s − 16·87-s + 32·93-s + 24·103-s + 20·109-s + 48·111-s + 12·113-s − 22·121-s + 127-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.377·7-s + 2·9-s + 0.917·19-s + 0.872·21-s + 6/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s − 1.97·37-s + 1.16·47-s + 1/7·49-s − 2.74·53-s − 2.11·57-s − 1.56·59-s − 0.755·63-s − 2.77·75-s − 4.11·81-s − 1.31·83-s − 1.71·87-s + 3.31·93-s + 2.36·103-s + 1.91·109-s + 4.55·111-s + 1.12·113-s − 2·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.27322067133513101779746201704, −9.590662754018899629342276160434, −8.870187268949863787955145092048, −8.605229056688940905430426859414, −7.48523560915642098619060234168, −7.22672629765934394974316502798, −6.34948564084072494191115955004, −6.27767574855681654030919817110, −5.57466416223045488855827312533, −4.90170998996837834018405216678, −4.88658861407619853654072156509, −3.61118272831897977078492843408, −2.87703403465824901690577931359, −1.30690906969982341922250798482, 0,
1.30690906969982341922250798482, 2.87703403465824901690577931359, 3.61118272831897977078492843408, 4.88658861407619853654072156509, 4.90170998996837834018405216678, 5.57466416223045488855827312533, 6.27767574855681654030919817110, 6.34948564084072494191115955004, 7.22672629765934394974316502798, 7.48523560915642098619060234168, 8.605229056688940905430426859414, 8.870187268949863787955145092048, 9.590662754018899629342276160434, 10.27322067133513101779746201704
