L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 3·9-s + 6·11-s − 8·13-s − 4·15-s − 2·17-s − 8·19-s + 2·21-s − 6·23-s + 2·25-s − 4·27-s + 4·29-s + 8·31-s − 12·33-s − 2·35-s + 8·37-s + 16·39-s − 6·41-s + 6·45-s + 12·47-s − 6·49-s + 4·51-s − 8·53-s + 12·55-s + 16·57-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 9-s + 1.80·11-s − 2.21·13-s − 1.03·15-s − 0.485·17-s − 1.83·19-s + 0.436·21-s − 1.25·23-s + 2/5·25-s − 0.769·27-s + 0.742·29-s + 1.43·31-s − 2.08·33-s − 0.338·35-s + 1.31·37-s + 2.56·39-s − 0.937·41-s + 0.894·45-s + 1.75·47-s − 6/7·49-s + 0.560·51-s − 1.09·53-s + 1.61·55-s + 2.11·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5242823665\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5242823665\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−19.1003840744, −17.9691621778, −17.6707519500, −17.1205967070, −16.8490958895, −16.6272610009, −15.4732516475, −15.1856008624, −14.1827177830, −14.1819882616, −13.1571731230, −12.3337462181, −12.3172568607, −11.5820474614, −10.8404426010, −9.96467191573, −9.84644221106, −9.09431466285, −8.09899069409, −6.96207355784, −6.42897107073, −6.05700480174, −4.80916909917, −4.25303028693, −2.24239796256,
2.24239796256, 4.25303028693, 4.80916909917, 6.05700480174, 6.42897107073, 6.96207355784, 8.09899069409, 9.09431466285, 9.84644221106, 9.96467191573, 10.8404426010, 11.5820474614, 12.3172568607, 12.3337462181, 13.1571731230, 14.1819882616, 14.1827177830, 15.1856008624, 15.4732516475, 16.6272610009, 16.8490958895, 17.1205967070, 17.6707519500, 17.9691621778, 19.1003840744