| L(s) = 1 | − 2-s − 4·3-s + 4-s − 4·5-s + 4·6-s − 3·7-s − 8-s + 7·9-s + 4·10-s − 4·11-s − 4·12-s − 4·13-s + 3·14-s + 16·15-s + 16-s − 4·17-s − 7·18-s − 4·20-s + 12·21-s + 4·22-s − 6·23-s + 4·24-s + 7·25-s + 4·26-s − 4·27-s − 3·28-s + 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 1.63·6-s − 1.13·7-s − 0.353·8-s + 7/3·9-s + 1.26·10-s − 1.20·11-s − 1.15·12-s − 1.10·13-s + 0.801·14-s + 4.13·15-s + 1/4·16-s − 0.970·17-s − 1.64·18-s − 0.894·20-s + 2.61·21-s + 0.852·22-s − 1.25·23-s + 0.816·24-s + 7/5·25-s + 0.784·26-s − 0.769·27-s − 0.566·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 1907 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 29 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 74 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T - 65 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 52 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 123 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 23 T + 3 p T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
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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
−16.6663770728, −16.0156219306, −15.8708779654, −15.5733748636, −15.0242315898, −14.2480261738, −13.3436116010, −12.7864692444, −12.2670515695, −12.0717864966, −11.6209805502, −11.0905606106, −10.7820988842, −10.1760116870, −9.81105534598, −8.92909993499, −8.13583138126, −7.66520019160, −7.11424039809, −6.44070593676, −6.09208051893, −5.17137598374, −4.72042419532, −3.79323740141, −2.69804366590, 0, 0,
2.69804366590, 3.79323740141, 4.72042419532, 5.17137598374, 6.09208051893, 6.44070593676, 7.11424039809, 7.66520019160, 8.13583138126, 8.92909993499, 9.81105534598, 10.1760116870, 10.7820988842, 11.0905606106, 11.6209805502, 12.0717864966, 12.2670515695, 12.7864692444, 13.3436116010, 14.2480261738, 15.0242315898, 15.5733748636, 15.8708779654, 16.0156219306, 16.6663770728
