| L(s) = 1 | − 4·7-s + 9-s + 8·11-s − 4·13-s − 4·17-s − 8·19-s − 8·23-s − 6·25-s + 8·29-s + 12·31-s + 4·37-s − 4·41-s + 8·43-s + 8·47-s + 2·49-s + 8·53-s + 4·61-s − 4·63-s − 8·71-s + 4·73-s − 32·77-s − 4·79-s + 81-s + 8·83-s + 4·89-s + 16·91-s − 12·97-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1/3·9-s + 2.41·11-s − 1.10·13-s − 0.970·17-s − 1.83·19-s − 1.66·23-s − 6/5·25-s + 1.48·29-s + 2.15·31-s + 0.657·37-s − 0.624·41-s + 1.21·43-s + 1.16·47-s + 2/7·49-s + 1.09·53-s + 0.512·61-s − 0.503·63-s − 0.949·71-s + 0.468·73-s − 3.64·77-s − 0.450·79-s + 1/9·81-s + 0.878·83-s + 0.423·89-s + 1.67·91-s − 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6315464353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6315464353\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−19.1551537949, −17.7368972379, −17.6707519500, −17.0027248656, −16.6272610009, −15.9210516795, −15.4732516475, −14.9200761308, −14.1819882616, −13.8007317307, −13.1414049036, −12.3172568607, −12.1054145849, −11.5820474614, −10.4491330419, −9.96467191573, −9.48869923766, −8.83988629202, −8.09899069409, −6.91469103627, −6.42897107073, −6.15267013016, −4.25303028693, −4.10625798014, −2.44821729992,
2.44821729992, 4.10625798014, 4.25303028693, 6.15267013016, 6.42897107073, 6.91469103627, 8.09899069409, 8.83988629202, 9.48869923766, 9.96467191573, 10.4491330419, 11.5820474614, 12.1054145849, 12.3172568607, 13.1414049036, 13.8007317307, 14.1819882616, 14.9200761308, 15.4732516475, 15.9210516795, 16.6272610009, 17.0027248656, 17.6707519500, 17.7368972379, 19.1551537949
