| L(s) = 1 | − 3·3-s + 5-s + 5·7-s + 3·9-s − 11-s − 4·13-s − 3·15-s − 6·17-s − 2·19-s − 15·21-s − 2·23-s − 4·25-s − 6·29-s + 3·33-s + 5·35-s − 5·37-s + 12·39-s − 11·41-s − 2·43-s + 3·45-s + 9·47-s + 7·49-s + 18·51-s + 5·53-s − 55-s + 6·57-s − 6·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 1.88·7-s + 9-s − 0.301·11-s − 1.10·13-s − 0.774·15-s − 1.45·17-s − 0.458·19-s − 3.27·21-s − 0.417·23-s − 4/5·25-s − 1.11·29-s + 0.522·33-s + 0.845·35-s − 0.821·37-s + 1.92·39-s − 1.71·41-s − 0.304·43-s + 0.447·45-s + 1.31·47-s + 49-s + 2.52·51-s + 0.686·53-s − 0.134·55-s + 0.794·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 96 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 70 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 134 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−14.9738155878, −14.7204503073, −14.0686953466, −13.6593245843, −13.2510637229, −12.5214143413, −12.1442343378, −11.6408791935, −11.3909452921, −11.0250760055, −10.5373609811, −10.1557270876, −9.46343208623, −8.82983576438, −8.28668932354, −7.77772302339, −7.12200507490, −6.64697882277, −5.90793613964, −5.47516114216, −4.98510030380, −4.66502880627, −3.84954696463, −2.29023093783, −1.79492186781, 0,
1.79492186781, 2.29023093783, 3.84954696463, 4.66502880627, 4.98510030380, 5.47516114216, 5.90793613964, 6.64697882277, 7.12200507490, 7.77772302339, 8.28668932354, 8.82983576438, 9.46343208623, 10.1557270876, 10.5373609811, 11.0250760055, 11.3909452921, 11.6408791935, 12.1442343378, 12.5214143413, 13.2510637229, 13.6593245843, 14.0686953466, 14.7204503073, 14.9738155878
