| L(s) = 1 | − 2-s − 4·3-s + 4-s − 2·5-s + 4·6-s + 7-s − 8-s + 6·9-s + 2·10-s − 4·11-s − 4·12-s − 10·13-s − 14-s + 8·15-s + 16-s + 2·17-s − 6·18-s − 4·19-s − 2·20-s − 4·21-s + 4·22-s − 4·23-s + 4·24-s − 6·25-s + 10·26-s + 4·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s + 0.377·7-s − 0.353·8-s + 2·9-s + 0.632·10-s − 1.20·11-s − 1.15·12-s − 2.77·13-s − 0.267·14-s + 2.06·15-s + 1/4·16-s + 0.485·17-s − 1.41·18-s − 0.917·19-s − 0.447·20-s − 0.872·21-s + 0.852·22-s − 0.834·23-s + 0.816·24-s − 6/5·25-s + 1.96·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−15.4514933430, −14.9728953745, −14.6077730657, −14.2451399800, −13.1881895766, −12.6417026271, −12.3052609901, −12.1063406379, −11.4418273496, −11.2313614144, −10.9409991928, −10.2459627166, −9.76554711946, −9.63516922865, −8.37116758932, −8.01003490126, −7.57571100089, −7.08238367024, −6.45217559209, −5.59673699857, −5.57928681743, −4.69839552642, −4.43201465917, −3.00626769255, −2.08688930207, 0, 0,
2.08688930207, 3.00626769255, 4.43201465917, 4.69839552642, 5.57928681743, 5.59673699857, 6.45217559209, 7.08238367024, 7.57571100089, 8.01003490126, 8.37116758932, 9.63516922865, 9.76554711946, 10.2459627166, 10.9409991928, 11.2313614144, 11.4418273496, 12.1063406379, 12.3052609901, 12.6417026271, 13.1881895766, 14.2451399800, 14.6077730657, 14.9728953745, 15.4514933430
