| L(s) = 1 | − 5-s + 4·7-s − 9-s − 4·11-s − 2·13-s − 6·17-s + 4·23-s + 25-s + 4·31-s − 4·35-s − 2·37-s − 8·41-s + 45-s + 4·47-s + 6·49-s − 2·53-s + 4·55-s − 4·63-s + 2·65-s − 4·71-s − 6·73-s − 16·77-s + 16·79-s + 81-s − 24·83-s + 6·85-s − 8·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1/3·9-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.834·23-s + 1/5·25-s + 0.718·31-s − 0.676·35-s − 0.328·37-s − 1.24·41-s + 0.149·45-s + 0.583·47-s + 6/7·49-s − 0.274·53-s + 0.539·55-s − 0.503·63-s + 0.248·65-s − 0.474·71-s − 0.702·73-s − 1.82·77-s + 1.80·79-s + 1/9·81-s − 2.63·83-s + 0.650·85-s − 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−13.2769174108, −12.9384804592, −12.5181834861, −11.8278605258, −11.7290538668, −11.2054870837, −10.8642673507, −10.5469113744, −10.0647873436, −9.49990579259, −8.82654513439, −8.58303937912, −8.20345448790, −7.74748308950, −7.27261909584, −6.90429655268, −6.25916617929, −5.56155617801, −5.06904341557, −4.73569653805, −4.35809392683, −3.51250280884, −2.70829653746, −2.29580112657, −1.39416876685, 0,
1.39416876685, 2.29580112657, 2.70829653746, 3.51250280884, 4.35809392683, 4.73569653805, 5.06904341557, 5.56155617801, 6.25916617929, 6.90429655268, 7.27261909584, 7.74748308950, 8.20345448790, 8.58303937912, 8.82654513439, 9.49990579259, 10.0647873436, 10.5469113744, 10.8642673507, 11.2054870837, 11.7290538668, 11.8278605258, 12.5181834861, 12.9384804592, 13.2769174108
