| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s − 2·16-s − 18·23-s − 4·27-s − 20·31-s + 36-s − 16·37-s − 24·47-s + 4·48-s − 19·49-s − 18·53-s − 12·59-s − 64-s − 4·67-s + 36·69-s − 4·81-s + 42·89-s − 18·92-s + 40·93-s − 26·97-s − 38·103-s − 4·108-s + 32·111-s + 12·113-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s − 1/2·16-s − 3.75·23-s − 0.769·27-s − 3.59·31-s + 1/6·36-s − 2.63·37-s − 3.50·47-s + 0.577·48-s − 2.71·49-s − 2.47·53-s − 1.56·59-s − 1/8·64-s − 0.488·67-s + 4.33·69-s − 4/9·81-s + 4.45·89-s − 1.87·92-s + 4.14·93-s − 2.63·97-s − 3.74·103-s − 0.384·108-s + 3.03·111-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - T^{2} + 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_{4}$ | \( ( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 19 T^{2} + 183 T^{4} + 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 213 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 29 T^{2} + 783 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 261 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 9 T + 61 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 19 T^{2} + 591 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 104 T^{2} + 5541 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 136 T^{2} + 8238 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 9 T + 121 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 79 T^{2} + 2571 T^{4} + 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 151 T^{2} + 15723 T^{4} + 151 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + T^{2} + 6051 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 281 T^{2} + 32883 T^{4} + 281 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 21 T + 283 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 13 T + 189 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.55221673002952835982412212628, −6.23668313702133705004103379402, −6.19373021199815489848921093292, −6.08781523983833724030976276601, −5.86745620294419884514625066866, −5.48534750619571270017557449227, −5.29513535885483983433532663765, −5.26032994689890223889942001943, −5.15304458986603126726249485262, −4.86279361492861530195259560296, −4.48804146063059028340765631951, −4.37584449596002252215931110429, −4.06891864224617506979556403757, −3.97300480364838044049344998481, −3.66632415427217356135060005273, −3.31237883974452543276890055131, −3.29850863774756428655976131414, −3.07563861641573480394259833739, −2.89593804408934521839801885803, −2.15226244202004724871599176430, −2.00838092118758381379881051074, −1.81861708543312161596572150773, −1.75629968365430245147695776899, −1.64361869270343592616466887494, −1.14197876920113854631443967665, 0, 0, 0, 0,
1.14197876920113854631443967665, 1.64361869270343592616466887494, 1.75629968365430245147695776899, 1.81861708543312161596572150773, 2.00838092118758381379881051074, 2.15226244202004724871599176430, 2.89593804408934521839801885803, 3.07563861641573480394259833739, 3.29850863774756428655976131414, 3.31237883974452543276890055131, 3.66632415427217356135060005273, 3.97300480364838044049344998481, 4.06891864224617506979556403757, 4.37584449596002252215931110429, 4.48804146063059028340765631951, 4.86279361492861530195259560296, 5.15304458986603126726249485262, 5.26032994689890223889942001943, 5.29513535885483983433532663765, 5.48534750619571270017557449227, 5.86745620294419884514625066866, 6.08781523983833724030976276601, 6.19373021199815489848921093292, 6.23668313702133705004103379402, 6.55221673002952835982412212628
