| L(s) = 1 | + 2-s + 2·3-s + 4-s − 4·5-s + 2·6-s + 2·7-s + 8-s + 9-s − 4·10-s + 4·11-s + 2·12-s − 2·13-s + 2·14-s − 8·15-s + 16-s + 6·17-s + 18-s − 2·19-s − 4·20-s + 4·21-s + 4·22-s + 2·23-s + 2·24-s + 6·25-s − 2·26-s + 2·27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 1.20·11-s + 0.577·12-s − 0.554·13-s + 0.534·14-s − 2.06·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.894·20-s + 0.872·21-s + 0.852·22-s + 0.417·23-s + 0.408·24-s + 6/5·25-s − 0.392·26-s + 0.384·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.120076109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.120076109\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−15.4150132989, −15.0017429576, −14.4486370331, −14.3734863297, −14.0650056929, −13.2789004951, −12.4946108502, −12.3882574259, −11.6151508324, −11.6097693908, −10.9885026417, −10.1555547780, −9.75290018913, −8.79659055688, −8.40848690901, −8.19782081364, −7.28048593381, −7.25038064773, −6.39357456655, −5.28069074518, −4.82665740673, −3.86057015395, −3.66834687648, −2.94857221125, −1.66549496948,
1.66549496948, 2.94857221125, 3.66834687648, 3.86057015395, 4.82665740673, 5.28069074518, 6.39357456655, 7.25038064773, 7.28048593381, 8.19782081364, 8.40848690901, 8.79659055688, 9.75290018913, 10.1555547780, 10.9885026417, 11.6097693908, 11.6151508324, 12.3882574259, 12.4946108502, 13.2789004951, 14.0650056929, 14.3734863297, 14.4486370331, 15.0017429576, 15.4150132989
