| L(s) = 1 | + 2-s + 2·5-s + 2·7-s + 8-s + 9-s + 2·10-s + 3·11-s − 13-s + 2·14-s − 16-s + 2·17-s + 18-s + 2·19-s + 3·22-s − 4·23-s − 26-s + 6·29-s − 2·31-s − 6·32-s + 2·34-s + 4·35-s − 4·37-s + 2·38-s + 2·40-s + 4·43-s + 2·45-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.894·5-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.904·11-s − 0.277·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.458·19-s + 0.639·22-s − 0.834·23-s − 0.196·26-s + 1.11·29-s − 0.359·31-s − 1.06·32-s + 0.342·34-s + 0.676·35-s − 0.657·37-s + 0.324·38-s + 0.316·40-s + 0.609·43-s + 0.298·45-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100017 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100017 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.965947142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.965947142\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11113 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 145 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T - 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 3 T - 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 136 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.0518361350, −13.8656130795, −13.2395745952, −12.8549014706, −12.1997779967, −12.1095886148, −11.4745340319, −10.9827784013, −10.5425709816, −10.0039615403, −9.62779128433, −9.14908147388, −8.63368259139, −8.07894874165, −7.38940224801, −7.15982625688, −6.33178291848, −5.96143923033, −5.36271924894, −4.77242007231, −4.38217375266, −3.76890945763, −2.92058135340, −1.95741762698, −1.39937843484,
1.39937843484, 1.95741762698, 2.92058135340, 3.76890945763, 4.38217375266, 4.77242007231, 5.36271924894, 5.96143923033, 6.33178291848, 7.15982625688, 7.38940224801, 8.07894874165, 8.63368259139, 9.14908147388, 9.62779128433, 10.0039615403, 10.5425709816, 10.9827784013, 11.4745340319, 12.1095886148, 12.1997779967, 12.8549014706, 13.2395745952, 13.8656130795, 14.0518361350
