| L(s) = 1 | − 2-s − 4·3-s − 4-s − 6·5-s + 4·6-s − 8·7-s + 3·8-s + 7·9-s + 6·10-s + 4·12-s − 10·13-s + 8·14-s + 24·15-s − 16-s + 2·17-s − 7·18-s − 2·19-s + 6·20-s + 32·21-s − 4·23-s − 12·24-s + 18·25-s + 10·26-s − 4·27-s + 8·28-s − 2·29-s − 24·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 1/2·4-s − 2.68·5-s + 1.63·6-s − 3.02·7-s + 1.06·8-s + 7/3·9-s + 1.89·10-s + 1.15·12-s − 2.77·13-s + 2.13·14-s + 6.19·15-s − 1/4·16-s + 0.485·17-s − 1.64·18-s − 0.458·19-s + 1.34·20-s + 6.98·21-s − 0.834·23-s − 2.44·24-s + 18/5·25-s + 1.96·26-s − 0.769·27-s + 1.51·28-s − 0.371·29-s − 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 83 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−15.6808225959, −14.9504044198, −14.8672048131, −13.8234941000, −13.2058537713, −12.6126147637, −12.3053459066, −12.2033077357, −11.7789043497, −11.4390202070, −10.5699724606, −10.1767663545, −10.0887048962, −9.45782766992, −8.81819836919, −8.11892799393, −7.50319728141, −6.97450299304, −6.91414715617, −6.17537190046, −5.31327024180, −4.96934813989, −4.19529103257, −3.65475901244, −3.01044735122, 0, 0, 0,
3.01044735122, 3.65475901244, 4.19529103257, 4.96934813989, 5.31327024180, 6.17537190046, 6.91414715617, 6.97450299304, 7.50319728141, 8.11892799393, 8.81819836919, 9.45782766992, 10.0887048962, 10.1767663545, 10.5699724606, 11.4390202070, 11.7789043497, 12.2033077357, 12.3053459066, 12.6126147637, 13.2058537713, 13.8234941000, 14.8672048131, 14.9504044198, 15.6808225959
