| L(s) = 1 | + 2·2-s + 5·5-s − 2·7-s − 4·8-s + 2·9-s + 10·10-s + 7·11-s − 4·13-s − 4·14-s − 4·16-s + 3·17-s + 4·18-s − 9·19-s + 14·22-s − 3·23-s + 11·25-s − 8·26-s + 29-s + 10·31-s + 6·34-s − 10·35-s + 9·37-s − 18·38-s − 20·40-s + 9·41-s + 10·45-s − 6·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.23·5-s − 0.755·7-s − 1.41·8-s + 2/3·9-s + 3.16·10-s + 2.11·11-s − 1.10·13-s − 1.06·14-s − 16-s + 0.727·17-s + 0.942·18-s − 2.06·19-s + 2.98·22-s − 0.625·23-s + 11/5·25-s − 1.56·26-s + 0.185·29-s + 1.79·31-s + 1.02·34-s − 1.69·35-s + 1.47·37-s − 2.91·38-s − 3.16·40-s + 1.40·41-s + 1.49·45-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79397 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79397 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.517578370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.517578370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 79397 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 106 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 78 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 15 T + 115 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 99 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 29 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T - 81 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
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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
−14.2337405922, −13.7730454754, −13.4819219811, −13.0951352938, −12.5857397555, −12.3651431866, −12.0598524348, −11.2516189196, −10.5045048714, −10.0504857383, −9.67090432153, −9.33865581265, −9.16400052349, −8.37503145759, −7.60012864442, −6.62822114287, −6.40980234893, −6.02542730213, −5.72981409926, −4.70607453449, −4.45390807779, −3.99685299058, −3.03335140682, −2.32410893419, −1.42853825160,
1.42853825160, 2.32410893419, 3.03335140682, 3.99685299058, 4.45390807779, 4.70607453449, 5.72981409926, 6.02542730213, 6.40980234893, 6.62822114287, 7.60012864442, 8.37503145759, 9.16400052349, 9.33865581265, 9.67090432153, 10.0504857383, 10.5045048714, 11.2516189196, 12.0598524348, 12.3651431866, 12.5857397555, 13.0951352938, 13.4819219811, 13.7730454754, 14.2337405922
