| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 4·7-s − 2·9-s + 2·10-s − 11-s − 2·12-s − 8·13-s + 8·14-s + 15-s − 4·16-s + 4·18-s − 2·20-s + 4·21-s + 2·22-s + 3·23-s + 25-s + 16·26-s + 5·27-s − 8·28-s + 9·29-s − 2·30-s − 5·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 1.51·7-s − 2/3·9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s − 2.21·13-s + 2.13·14-s + 0.258·15-s − 16-s + 0.942·18-s − 0.447·20-s + 0.872·21-s + 0.426·22-s + 0.625·23-s + 1/5·25-s + 3.13·26-s + 0.962·27-s − 1.51·28-s + 1.67·29-s − 0.365·30-s − 0.898·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 63 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 3 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 135 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 144 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T - 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T - 107 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 28 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.6221076576, −12.3061342804, −12.0451286885, −11.6913171409, −10.9898416160, −10.7714601560, −10.3826906530, −10.0171233944, −9.51788459260, −9.35990479244, −8.87059089309, −8.45084139886, −7.89282626371, −7.49843324936, −7.07526239426, −6.80360965541, −6.26669224094, −5.77979976753, −5.05502674544, −4.80627653320, −4.15715616608, −3.27711290335, −2.70111978168, −2.47535212109, −1.19827611875, 0, 0,
1.19827611875, 2.47535212109, 2.70111978168, 3.27711290335, 4.15715616608, 4.80627653320, 5.05502674544, 5.77979976753, 6.26669224094, 6.80360965541, 7.07526239426, 7.49843324936, 7.89282626371, 8.45084139886, 8.87059089309, 9.35990479244, 9.51788459260, 10.0171233944, 10.3826906530, 10.7714601560, 10.9898416160, 11.6913171409, 12.0451286885, 12.3061342804, 12.6221076576
