| L(s) = 1 | + 2-s + 3·5-s − 7-s + 8-s − 2·9-s + 3·10-s − 5·13-s − 14-s − 16-s − 3·17-s − 2·18-s − 6·19-s + 3·23-s + 5·25-s − 5·26-s + 4·29-s + 6·31-s − 6·32-s − 3·34-s − 3·35-s + 37-s − 6·38-s + 3·40-s + 10·41-s + 43-s − 6·45-s + 3·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·5-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 1.38·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.37·19-s + 0.625·23-s + 25-s − 0.980·26-s + 0.742·29-s + 1.07·31-s − 1.06·32-s − 0.514·34-s − 0.507·35-s + 0.164·37-s − 0.973·38-s + 0.474·40-s + 1.56·41-s + 0.152·43-s − 0.894·45-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7657 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7657 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.348420793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.348420793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 5 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 55 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 21 T + 214 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 46 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T - 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 123 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−17.0965787465, −16.5379291242, −15.9317463024, −15.2431813352, −14.7923777323, −14.3185482288, −13.8206599415, −13.4936626990, −12.9931070878, −12.4731265818, −12.0306568372, −11.1491262382, −10.6073811297, −10.2356391628, −9.39869725078, −9.09707893967, −8.43641200844, −7.45233075474, −6.82646823042, −6.13794741333, −5.65339994729, −4.66910573323, −4.41109581740, −2.82988953205, −2.23654548682,
2.23654548682, 2.82988953205, 4.41109581740, 4.66910573323, 5.65339994729, 6.13794741333, 6.82646823042, 7.45233075474, 8.43641200844, 9.09707893967, 9.39869725078, 10.2356391628, 10.6073811297, 11.1491262382, 12.0306568372, 12.4731265818, 12.9931070878, 13.4936626990, 13.8206599415, 14.3185482288, 14.7923777323, 15.2431813352, 15.9317463024, 16.5379291242, 17.0965787465
