| L(s) = 1 | − 3-s + 4-s − 5-s + 9-s + 2·11-s − 12-s + 4·13-s + 15-s − 3·16-s + 4·17-s + 6·19-s − 20-s − 4·23-s − 2·25-s − 27-s − 10·29-s + 2·31-s − 2·33-s + 36-s − 4·39-s − 4·41-s − 12·43-s + 2·44-s − 45-s − 8·47-s + 3·48-s − 7·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 1.10·13-s + 0.258·15-s − 3/4·16-s + 0.970·17-s + 1.37·19-s − 0.223·20-s − 0.834·23-s − 2/5·25-s − 0.192·27-s − 1.85·29-s + 0.359·31-s − 0.348·33-s + 1/6·36-s − 0.640·39-s − 0.624·41-s − 1.82·43-s + 0.301·44-s − 0.149·45-s − 1.16·47-s + 0.433·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8295318623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8295318623\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T - 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−16.9208924383, −16.4630874841, −16.2397566666, −15.7964294299, −15.1502143415, −14.7468425365, −14.0699434937, −13.3660582775, −13.2200131717, −12.2105321664, −11.7512559818, −11.4769745016, −11.1194206883, −10.1374491769, −9.83983589356, −9.07297702439, −8.30842730464, −7.71581474038, −7.03712971143, −6.45370608558, −5.73222474399, −5.08259781538, −3.92784521913, −3.37169891372, −1.66658984670,
1.66658984670, 3.37169891372, 3.92784521913, 5.08259781538, 5.73222474399, 6.45370608558, 7.03712971143, 7.71581474038, 8.30842730464, 9.07297702439, 9.83983589356, 10.1374491769, 11.1194206883, 11.4769745016, 11.7512559818, 12.2105321664, 13.2200131717, 13.3660582775, 14.0699434937, 14.7468425365, 15.1502143415, 15.7964294299, 16.2397566666, 16.4630874841, 16.9208924383
