| L(s) = 1 | − 2·3-s − 5-s − 3·7-s + 4·9-s + 11-s − 13-s + 2·15-s − 4·19-s + 6·21-s − 4·23-s − 3·25-s − 5·27-s − 8·29-s − 4·31-s − 2·33-s + 3·35-s + 6·37-s + 2·39-s − 3·41-s − 2·43-s − 4·45-s − 6·49-s + 2·53-s − 55-s + 8·57-s − 13·59-s + 15·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 1.13·7-s + 4/3·9-s + 0.301·11-s − 0.277·13-s + 0.516·15-s − 0.917·19-s + 1.30·21-s − 0.834·23-s − 3/5·25-s − 0.962·27-s − 1.48·29-s − 0.718·31-s − 0.348·33-s + 0.507·35-s + 0.986·37-s + 0.320·39-s − 0.468·41-s − 0.304·43-s − 0.596·45-s − 6/7·49-s + 0.274·53-s − 0.134·55-s + 1.05·57-s − 1.69·59-s + 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 199 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 20 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 91 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15 T + 155 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 82 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 67 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T - 72 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 104 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−16.1404550378, −15.6388066614, −15.2890537926, −14.6750534256, −14.2557130493, −13.3436620850, −12.9413915418, −12.8315285879, −12.0788999028, −11.6978951664, −11.2835417244, −10.6146799076, −10.2030484960, −9.62236302138, −9.27070567864, −8.44151582398, −7.65598748015, −7.24661997849, −6.50631072592, −6.13912823813, −5.55090451960, −4.63725270495, −4.03824216918, −3.34311662490, −1.92162121792, 0,
1.92162121792, 3.34311662490, 4.03824216918, 4.63725270495, 5.55090451960, 6.13912823813, 6.50631072592, 7.24661997849, 7.65598748015, 8.44151582398, 9.27070567864, 9.62236302138, 10.2030484960, 10.6146799076, 11.2835417244, 11.6978951664, 12.0788999028, 12.8315285879, 12.9413915418, 13.3436620850, 14.2557130493, 14.6750534256, 15.2890537926, 15.6388066614, 16.1404550378
