| L(s) = 1 | + 2-s + 3-s + 3·5-s + 6-s + 8-s − 2·9-s + 3·10-s + 11-s − 13-s + 3·15-s − 16-s + 4·17-s − 2·18-s + 19-s + 22-s − 23-s + 24-s − 25-s − 26-s − 5·27-s − 7·29-s + 3·30-s − 6·32-s + 33-s + 4·34-s + 4·37-s + 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1.34·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.301·11-s − 0.277·13-s + 0.774·15-s − 1/4·16-s + 0.970·17-s − 0.471·18-s + 0.229·19-s + 0.213·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.962·27-s − 1.29·29-s + 0.547·30-s − 1.06·32-s + 0.174·33-s + 0.685·34-s + 0.657·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30429 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30429 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.410045581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.410045581\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 4 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13 T + 121 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T - 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 119 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 75 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 112 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 105 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−14.9472643100, −14.7175882238, −14.1571810948, −13.9305145983, −13.5195043095, −13.1435781302, −12.7582255593, −11.9736121309, −11.5753937414, −11.1391771846, −10.2978559976, −9.95116259809, −9.48531051657, −9.06560763399, −8.39672189377, −7.86368969562, −7.22014132365, −6.57522117952, −5.84658323082, −5.43697187224, −4.99493406358, −3.93056615010, −3.46577220372, −2.40907134981, −1.78922414394,
1.78922414394, 2.40907134981, 3.46577220372, 3.93056615010, 4.99493406358, 5.43697187224, 5.84658323082, 6.57522117952, 7.22014132365, 7.86368969562, 8.39672189377, 9.06560763399, 9.48531051657, 9.95116259809, 10.2978559976, 11.1391771846, 11.5753937414, 11.9736121309, 12.7582255593, 13.1435781302, 13.5195043095, 13.9305145983, 14.1571810948, 14.7175882238, 14.9472643100
