L(s) = 1 | − 2-s − 4-s + 3·8-s + 2·9-s + 4·13-s − 16-s + 4·17-s − 2·18-s − 6·25-s − 4·26-s − 12·29-s − 5·32-s − 4·34-s − 2·36-s − 16·37-s + 12·41-s + 2·49-s + 6·50-s − 4·52-s + 8·53-s + 12·58-s + 4·61-s + 7·64-s − 4·68-s + 6·72-s − 12·73-s + 16·74-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s + 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.471·18-s − 6/5·25-s − 0.784·26-s − 2.22·29-s − 0.883·32-s − 0.685·34-s − 1/3·36-s − 2.63·37-s + 1.87·41-s + 2/7·49-s + 0.848·50-s − 0.554·52-s + 1.09·53-s + 1.57·58-s + 0.512·61-s + 7/8·64-s − 0.485·68-s + 0.707·72-s − 1.40·73-s + 1.85·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5440100461\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5440100461\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 257 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−12.55113911913291432504173264338, −11.85008133515623933385800153806, −11.05139471712736036943594714001, −10.67459149203692243797489835816, −9.861251387498423866382206694842, −9.610791028647234859608006801501, −8.765669568456885891616139520575, −8.371288656108001150737374927218, −7.39835362300314054472468513894, −7.24477666504451448204354780334, −5.87545876078630608591993295901, −5.41426981374372084835653925495, −4.16331826282344293868388099851, −3.63582255783349937241895391884, −1.64408879948474926792686178522,
1.64408879948474926792686178522, 3.63582255783349937241895391884, 4.16331826282344293868388099851, 5.41426981374372084835653925495, 5.87545876078630608591993295901, 7.24477666504451448204354780334, 7.39835362300314054472468513894, 8.371288656108001150737374927218, 8.765669568456885891616139520575, 9.610791028647234859608006801501, 9.861251387498423866382206694842, 10.67459149203692243797489835816, 11.05139471712736036943594714001, 11.85008133515623933385800153806, 12.55113911913291432504173264338