| L(s) = 1 | − 2-s − 3·3-s − 4-s − 3·5-s + 3·6-s − 2·7-s + 8-s + 4·9-s + 3·10-s − 6·11-s + 3·12-s − 3·13-s + 2·14-s + 9·15-s − 16-s − 2·17-s − 4·18-s − 8·19-s + 3·20-s + 6·21-s + 6·22-s − 2·23-s − 3·24-s + 3·26-s − 6·27-s + 2·28-s − 8·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 4/3·9-s + 0.948·10-s − 1.80·11-s + 0.866·12-s − 0.832·13-s + 0.534·14-s + 2.32·15-s − 1/4·16-s − 0.485·17-s − 0.942·18-s − 1.83·19-s + 0.670·20-s + 1.30·21-s + 1.27·22-s − 0.417·23-s − 0.612·24-s + 0.588·26-s − 1.15·27-s + 0.377·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 438869 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 438869 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 438869 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 515 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 13 T + 118 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T - 94 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 125 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 130 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−13.2717994866, −12.8477676366, −12.3588149021, −12.2293850522, −11.6336710723, −11.1865344764, −11.0175972351, −10.6315682964, −10.0892159299, −9.74548935752, −9.43467096563, −8.68742504409, −8.30430360917, −7.95542741678, −7.52952174514, −6.98500271793, −6.50905463274, −6.12308457194, −5.39765796460, −5.14527260024, −4.66170111454, −4.00624531870, −3.64909755483, −2.66493687136, −1.93831896812, 0, 0, 0,
1.93831896812, 2.66493687136, 3.64909755483, 4.00624531870, 4.66170111454, 5.14527260024, 5.39765796460, 6.12308457194, 6.50905463274, 6.98500271793, 7.52952174514, 7.95542741678, 8.30430360917, 8.68742504409, 9.43467096563, 9.74548935752, 10.0892159299, 10.6315682964, 11.0175972351, 11.1865344764, 11.6336710723, 12.2293850522, 12.3588149021, 12.8477676366, 13.2717994866
