| L(s) = 1 | − 2-s − 3·3-s − 4·5-s + 3·6-s − 2·7-s + 8-s + 4·9-s + 4·10-s − 4·11-s − 2·13-s + 2·14-s + 12·15-s − 16-s − 7·17-s − 4·18-s − 4·19-s + 6·21-s + 4·22-s − 6·23-s − 3·24-s + 6·25-s + 2·26-s − 4·29-s − 12·30-s + 12·33-s + 7·34-s + 8·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1.78·5-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 4/3·9-s + 1.26·10-s − 1.20·11-s − 0.554·13-s + 0.534·14-s + 3.09·15-s − 1/4·16-s − 1.69·17-s − 0.942·18-s − 0.917·19-s + 1.30·21-s + 0.852·22-s − 1.25·23-s − 0.612·24-s + 6/5·25-s + 0.392·26-s − 0.742·29-s − 2.19·30-s + 2.08·33-s + 1.20·34-s + 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23412 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23412 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 1951 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 69 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 36 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T - 26 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 94 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 178 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−15.9858091546, −15.7660978324, −15.5162066832, −14.8776989578, −14.3103279993, −13.2946730269, −12.9792613187, −12.7108227470, −11.9475409335, −11.6326420127, −11.2798479827, −10.8087394266, −10.3119364930, −9.92572841332, −9.09759596768, −8.51930800197, −7.89749552888, −7.54722966096, −6.85653095774, −6.29557904309, −5.74884230273, −4.74633861972, −4.49122779233, −3.67433048406, −2.41916134265, 0, 0,
2.41916134265, 3.67433048406, 4.49122779233, 4.74633861972, 5.74884230273, 6.29557904309, 6.85653095774, 7.54722966096, 7.89749552888, 8.51930800197, 9.09759596768, 9.92572841332, 10.3119364930, 10.8087394266, 11.2798479827, 11.6326420127, 11.9475409335, 12.7108227470, 12.9792613187, 13.2946730269, 14.3103279993, 14.8776989578, 15.5162066832, 15.7660978324, 15.9858091546
