| L(s) = 1 | − 2·2-s − 3·3-s − 3·5-s + 6·6-s − 4·7-s + 4·8-s + 2·9-s + 6·10-s − 3·11-s − 4·13-s + 8·14-s + 9·15-s − 4·16-s + 17-s − 4·18-s − 19-s + 12·21-s + 6·22-s − 5·23-s − 12·24-s + 3·25-s + 8·26-s + 6·27-s − 18·30-s − 4·31-s + 9·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s − 1.34·5-s + 2.44·6-s − 1.51·7-s + 1.41·8-s + 2/3·9-s + 1.89·10-s − 0.904·11-s − 1.10·13-s + 2.13·14-s + 2.32·15-s − 16-s + 0.242·17-s − 0.942·18-s − 0.229·19-s + 2.61·21-s + 1.27·22-s − 1.04·23-s − 2.44·24-s + 3/5·25-s + 1.56·26-s + 1.15·27-s − 3.28·30-s − 0.718·31-s + 1.56·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7403 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7403 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 673 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 34 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 53 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 13 T + 99 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 50 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 151 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 12 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 115 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 19 T + 211 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
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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
−17.4559393510, −17.0591084608, −16.6405598525, −16.2058609906, −15.8544515950, −15.4023072398, −14.5678037514, −13.8935484855, −13.2314131707, −12.4731173251, −12.2614139019, −11.8837541321, −10.9123117276, −10.7548483610, −10.1749910610, −9.45791699503, −9.21450849734, −8.19944056573, −7.81924177949, −7.22934754619, −6.37373482028, −5.73955853749, −4.95417473444, −4.21627736384, −3.06882015920, 0, 0,
3.06882015920, 4.21627736384, 4.95417473444, 5.73955853749, 6.37373482028, 7.22934754619, 7.81924177949, 8.19944056573, 9.21450849734, 9.45791699503, 10.1749910610, 10.7548483610, 10.9123117276, 11.8837541321, 12.2614139019, 12.4731173251, 13.2314131707, 13.8935484855, 14.5678037514, 15.4023072398, 15.8544515950, 16.2058609906, 16.6405598525, 17.0591084608, 17.4559393510
