| L(s) = 1 | + 2·2-s − 3-s + 2·5-s − 2·6-s − 4·8-s + 9-s + 4·10-s − 2·11-s − 8·13-s − 2·15-s − 4·16-s + 2·18-s − 8·19-s − 4·22-s + 4·24-s − 6·25-s − 16·26-s − 27-s + 4·29-s − 4·30-s − 16·31-s + 2·33-s + 2·37-s − 16·38-s + 8·39-s − 8·40-s + 10·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 0.894·5-s − 0.816·6-s − 1.41·8-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 2.21·13-s − 0.516·15-s − 16-s + 0.471·18-s − 1.83·19-s − 0.852·22-s + 0.816·24-s − 6/5·25-s − 3.13·26-s − 0.192·27-s + 0.742·29-s − 0.730·30-s − 2.87·31-s + 0.348·33-s + 0.328·37-s − 2.59·38-s + 1.28·39-s − 1.26·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
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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
−14.5943800716, −14.2932398855, −13.8877947999, −13.0504144744, −13.0238540988, −12.6469887717, −12.4086664145, −11.6591047367, −11.3016282861, −10.4434096149, −10.2813603514, −9.75101739390, −9.09256398356, −8.94737293191, −7.99137209561, −7.29485403541, −7.07511528959, −6.01621490935, −5.72591573809, −5.40855511243, −4.78354615731, −4.18739539623, −3.87851392966, −2.47062022888, −2.26817487862, 0,
2.26817487862, 2.47062022888, 3.87851392966, 4.18739539623, 4.78354615731, 5.40855511243, 5.72591573809, 6.01621490935, 7.07511528959, 7.29485403541, 7.99137209561, 8.94737293191, 9.09256398356, 9.75101739390, 10.2813603514, 10.4434096149, 11.3016282861, 11.6591047367, 12.4086664145, 12.6469887717, 13.0238540988, 13.0504144744, 13.8877947999, 14.2932398855, 14.5943800716
