| L(s) = 1 | − 2-s + 4-s − 8-s − 8·13-s + 16-s − 12·17-s − 10·25-s + 8·26-s − 12·29-s − 32-s + 12·34-s − 8·37-s − 12·41-s + 2·49-s + 10·50-s − 8·52-s − 12·53-s + 12·58-s + 28·61-s + 64-s − 12·68-s + 28·73-s + 8·74-s + 12·82-s + 12·89-s − 20·97-s − 2·98-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2.21·13-s + 1/4·16-s − 2.91·17-s − 2·25-s + 1.56·26-s − 2.22·29-s − 0.176·32-s + 2.05·34-s − 1.31·37-s − 1.87·41-s + 2/7·49-s + 1.41·50-s − 1.10·52-s − 1.64·53-s + 1.57·58-s + 3.58·61-s + 1/8·64-s − 1.45·68-s + 3.27·73-s + 0.929·74-s + 1.32·82-s + 1.27·89-s − 2.03·97-s − 0.202·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.72795819597559924442133345234, −7.10678703287277123493279316153, −7.06756557503064388001592546481, −6.59547283064248335138694127148, −6.07507983377375758209992822317, −5.21403289865941834256074312934, −5.21083603966480863402338802043, −4.54790713149621293279441185278, −3.81156610301665935267786910207, −3.58677867797817326821260752741, −2.34092707995523044468378525917, −2.27147129954073815802695201168, −1.81801100280978906223857531826, 0, 0,
1.81801100280978906223857531826, 2.27147129954073815802695201168, 2.34092707995523044468378525917, 3.58677867797817326821260752741, 3.81156610301665935267786910207, 4.54790713149621293279441185278, 5.21083603966480863402338802043, 5.21403289865941834256074312934, 6.07507983377375758209992822317, 6.59547283064248335138694127148, 7.06756557503064388001592546481, 7.10678703287277123493279316153, 7.72795819597559924442133345234
