| L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s − 2·7-s − 8-s + 3·9-s − 10-s + 5·11-s − 2·12-s − 13-s − 2·14-s − 2·15-s − 16-s − 7·17-s + 3·18-s + 5·19-s + 20-s − 4·21-s + 5·22-s − 3·23-s − 2·24-s − 7·25-s − 26-s + 4·27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s + 9-s − 0.316·10-s + 1.50·11-s − 0.577·12-s − 0.277·13-s − 0.534·14-s − 0.516·15-s − 1/4·16-s − 1.69·17-s + 0.707·18-s + 1.14·19-s + 0.223·20-s − 0.872·21-s + 1.06·22-s − 0.625·23-s − 0.408·24-s − 7/5·25-s − 0.196·26-s + 0.769·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419285618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419285618\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−17.1598241820, −16.0578979019, −15.8734598001, −15.4839697445, −14.6332336925, −14.5135323404, −13.8080910134, −13.5051076814, −13.1937925053, −12.5213111167, −11.8472774623, −11.5559508175, −10.6780068798, −9.72466121073, −9.56507493454, −8.78186775993, −8.67236519668, −7.41845373611, −7.25047783803, −6.33620415860, −5.54552838750, −4.27441191084, −4.13559084051, −3.41109322659, −2.17139186064,
2.17139186064, 3.41109322659, 4.13559084051, 4.27441191084, 5.54552838750, 6.33620415860, 7.25047783803, 7.41845373611, 8.67236519668, 8.78186775993, 9.56507493454, 9.72466121073, 10.6780068798, 11.5559508175, 11.8472774623, 12.5213111167, 13.1937925053, 13.5051076814, 13.8080910134, 14.5135323404, 14.6332336925, 15.4839697445, 15.8734598001, 16.0578979019, 17.1598241820
