| L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s + 4·7-s + 3·8-s − 2·9-s − 2·10-s − 8·11-s + 12-s − 4·13-s − 4·14-s − 2·15-s − 16-s + 4·17-s + 2·18-s − 2·20-s − 4·21-s + 8·22-s − 4·23-s − 3·24-s + 3·25-s + 4·26-s + 2·27-s − 4·28-s − 4·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s + 1.06·8-s − 2/3·9-s − 0.632·10-s − 2.41·11-s + 0.288·12-s − 1.10·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.970·17-s + 0.471·18-s − 0.447·20-s − 0.872·21-s + 1.70·22-s − 0.834·23-s − 0.612·24-s + 3/5·25-s + 0.784·26-s + 0.384·27-s − 0.755·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3534607042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3534607042\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−19.3267648175, −19.1104624590, −18.1289881241, −18.1058537571, −17.5285223428, −17.2359534775, −16.6286476681, −15.9590260302, −15.1635791600, −14.4238186271, −14.0069258505, −13.5040682070, −12.6461787614, −12.1305048748, −11.1132149361, −10.6789224512, −10.1224683900, −9.52345167581, −8.52159014269, −7.75327442993, −7.66488013442, −5.84906736543, −5.23920392625, −4.76905661120, −2.39961978182,
2.39961978182, 4.76905661120, 5.23920392625, 5.84906736543, 7.66488013442, 7.75327442993, 8.52159014269, 9.52345167581, 10.1224683900, 10.6789224512, 11.1132149361, 12.1305048748, 12.6461787614, 13.5040682070, 14.0069258505, 14.4238186271, 15.1635791600, 15.9590260302, 16.6286476681, 17.2359534775, 17.5285223428, 18.1058537571, 18.1289881241, 19.1104624590, 19.3267648175
