| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s + 13-s + 16-s − 9·17-s + 3·20-s − 25-s + 26-s − 3·29-s + 32-s − 9·34-s + 4·37-s + 3·40-s − 6·41-s + 5·49-s − 50-s + 52-s − 9·53-s − 3·58-s + 7·61-s + 64-s + 3·65-s − 9·68-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 0.277·13-s + 1/4·16-s − 2.18·17-s + 0.670·20-s − 1/5·25-s + 0.196·26-s − 0.557·29-s + 0.176·32-s − 1.54·34-s + 0.657·37-s + 0.474·40-s − 0.937·41-s + 5/7·49-s − 0.141·50-s + 0.138·52-s − 1.23·53-s − 0.393·58-s + 0.896·61-s + 1/8·64-s + 0.372·65-s − 1.09·68-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.098270288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.098270288\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.86940989499473516200713839570, −10.22043364588793210856214470437, −9.691557136983968005336218844058, −9.232878910791911890735346680263, −8.665302057851250823565870445803, −8.045763775643920230789477152053, −7.19688828085222599171813160762, −6.67774560065291789857386110724, −6.11739323729356050184410284077, −5.72497144086868024570570088378, −4.92219463714602583925063672349, −4.34794453190773667034605855265, −3.49635674077201442896105115551, −2.39198150197047007970050424731, −1.87072905073531816574687630162,
1.87072905073531816574687630162, 2.39198150197047007970050424731, 3.49635674077201442896105115551, 4.34794453190773667034605855265, 4.92219463714602583925063672349, 5.72497144086868024570570088378, 6.11739323729356050184410284077, 6.67774560065291789857386110724, 7.19688828085222599171813160762, 8.045763775643920230789477152053, 8.665302057851250823565870445803, 9.232878910791911890735346680263, 9.691557136983968005336218844058, 10.22043364588793210856214470437, 10.86940989499473516200713839570
