| L(s) = 1 | + 2·2-s + 2·4-s + 6·5-s + 12·10-s + 4·13-s − 4·16-s + 2·17-s + 12·20-s + 17·25-s + 8·26-s + 4·29-s − 8·32-s + 4·34-s + 11·49-s + 34·50-s + 8·52-s − 20·53-s + 8·58-s − 2·61-s − 8·64-s + 24·65-s + 4·68-s − 22·73-s − 24·80-s + 12·85-s + 12·89-s − 20·97-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 2.68·5-s + 3.79·10-s + 1.10·13-s − 16-s + 0.485·17-s + 2.68·20-s + 17/5·25-s + 1.56·26-s + 0.742·29-s − 1.41·32-s + 0.685·34-s + 11/7·49-s + 4.80·50-s + 1.10·52-s − 2.74·53-s + 1.05·58-s − 0.256·61-s − 64-s + 2.97·65-s + 0.485·68-s − 2.57·73-s − 2.68·80-s + 1.30·85-s + 1.27·89-s − 2.03·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.104858821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.104858821\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 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
−8.742321760001961651389979974377, −8.160007849857414045140648086964, −7.43632371515983923998772054443, −6.85073886350288365363080648917, −6.29636417830616150950587212845, −6.12241751609850681792173625996, −5.80065514036351948381475791738, −5.34668146252658900228619106104, −4.91760568249510581017872290537, −4.32680757840779686690441622638, −3.67840796781373175574076394185, −2.94683125587980903660881898164, −2.61687998381550726785939862479, −1.81200784323266148009001357293, −1.34947676285250801422046885118,
1.34947676285250801422046885118, 1.81200784323266148009001357293, 2.61687998381550726785939862479, 2.94683125587980903660881898164, 3.67840796781373175574076394185, 4.32680757840779686690441622638, 4.91760568249510581017872290537, 5.34668146252658900228619106104, 5.80065514036351948381475791738, 6.12241751609850681792173625996, 6.29636417830616150950587212845, 6.85073886350288365363080648917, 7.43632371515983923998772054443, 8.160007849857414045140648086964, 8.742321760001961651389979974377
