| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 2·7-s − 8-s + 10-s − 6·11-s − 12-s + 2·13-s − 2·14-s + 15-s − 16-s + 2·17-s − 8·19-s − 20-s − 2·21-s − 6·22-s + 6·23-s − 24-s − 3·25-s + 2·26-s − 4·27-s + 2·28-s − 9·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 1.80·11-s − 0.288·12-s + 0.554·13-s − 0.534·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s − 1.83·19-s − 0.223·20-s − 0.436·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s − 3/5·25-s + 0.392·26-s − 0.769·27-s + 0.377·28-s − 1.67·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100059 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100059 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 33353 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 318 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13 T + 92 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 23 T + 222 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T - 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T - 60 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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
−14.1107792691, −13.7812350613, −13.3231079242, −13.0440158024, −12.8096626851, −12.4840975797, −11.7463481371, −10.9714073128, −10.6661034013, −10.4160921867, −9.79903551305, −9.14145341587, −8.78930442654, −8.66833536352, −7.71525753824, −7.30997909569, −6.95712838320, −5.89245636238, −5.59953081334, −5.35550910915, −4.40128267248, −3.89878868563, −3.40050715999, −2.53021885371, −1.99696640153, 0,
1.99696640153, 2.53021885371, 3.40050715999, 3.89878868563, 4.40128267248, 5.35550910915, 5.59953081334, 5.89245636238, 6.95712838320, 7.30997909569, 7.71525753824, 8.66833536352, 8.78930442654, 9.14145341587, 9.79903551305, 10.4160921867, 10.6661034013, 10.9714073128, 11.7463481371, 12.4840975797, 12.8096626851, 13.0440158024, 13.3231079242, 13.7812350613, 14.1107792691
