| L(s) = 1 | − 2-s + 4-s − 6·5-s − 6·7-s − 8-s + 6·10-s − 6·11-s + 4·13-s + 6·14-s + 16-s − 6·19-s − 6·20-s + 6·22-s + 6·23-s + 17·25-s − 4·26-s − 6·28-s + 6·29-s − 32-s + 36·35-s + 4·37-s + 6·38-s + 6·40-s + 12·41-s + 18·43-s − 6·44-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 2.68·5-s − 2.26·7-s − 0.353·8-s + 1.89·10-s − 1.80·11-s + 1.10·13-s + 1.60·14-s + 1/4·16-s − 1.37·19-s − 1.34·20-s + 1.27·22-s + 1.25·23-s + 17/5·25-s − 0.784·26-s − 1.13·28-s + 1.11·29-s − 0.176·32-s + 6.08·35-s + 0.657·37-s + 0.973·38-s + 0.948·40-s + 1.87·41-s + 2.74·43-s − 0.904·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−13.0510387346, −12.9974332898, −12.8940320235, −12.3270545018, −11.9542695681, −11.2891674053, −10.9202617439, −10.7854248521, −10.3183796818, −9.63897790310, −9.24737695522, −8.70031674519, −8.27894837570, −7.91644205358, −7.46771296380, −7.16277862973, −6.34883610203, −6.29261335709, −5.47820435939, −4.39411102666, −4.25638665319, −3.48342362060, −2.94651413140, −2.72095996214, −0.700574619211, 0,
0.700574619211, 2.72095996214, 2.94651413140, 3.48342362060, 4.25638665319, 4.39411102666, 5.47820435939, 6.29261335709, 6.34883610203, 7.16277862973, 7.46771296380, 7.91644205358, 8.27894837570, 8.70031674519, 9.24737695522, 9.63897790310, 10.3183796818, 10.7854248521, 10.9202617439, 11.2891674053, 11.9542695681, 12.3270545018, 12.8940320235, 12.9974332898, 13.0510387346
