| L(s) = 1 | − 2·2-s + 4-s − 2·7-s − 3·9-s − 4·11-s + 13-s + 4·14-s + 16-s − 5·17-s + 6·18-s − 19-s + 8·22-s + 4·23-s + 25-s − 2·26-s − 2·28-s − 4·29-s − 10·31-s + 2·32-s + 10·34-s − 3·36-s − 9·37-s + 2·38-s + 9·41-s + 9·43-s − 4·44-s − 8·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.755·7-s − 9-s − 1.20·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 1.21·17-s + 1.41·18-s − 0.229·19-s + 1.70·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.377·28-s − 0.742·29-s − 1.79·31-s + 0.353·32-s + 1.71·34-s − 1/2·36-s − 1.47·37-s + 0.324·38-s + 1.40·41-s + 1.37·43-s − 0.603·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 713293 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 713293 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 14557 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 203 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 76 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 39 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 51 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 66 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T - 75 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 114 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 144 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−12.7897883942, −12.3357594398, −12.0446547021, −11.2284732774, −10.8983247720, −10.8464451383, −10.4764128276, −9.76422511053, −9.36814675978, −9.16588675197, −8.80945029543, −8.51190749953, −8.01071143064, −7.44652885205, −7.26108782263, −6.61532618611, −6.09261839082, −5.59185091576, −5.37152297549, −4.55841335298, −4.01617771707, −3.32150939569, −2.71943105861, −2.37075479168, −1.31438116307, 0, 0,
1.31438116307, 2.37075479168, 2.71943105861, 3.32150939569, 4.01617771707, 4.55841335298, 5.37152297549, 5.59185091576, 6.09261839082, 6.61532618611, 7.26108782263, 7.44652885205, 8.01071143064, 8.51190749953, 8.80945029543, 9.16588675197, 9.36814675978, 9.76422511053, 10.4764128276, 10.8464451383, 10.8983247720, 11.2284732774, 12.0446547021, 12.3357594398, 12.7897883942
