| L(s) = 1 | − 3-s + 4-s − 4·5-s + 9-s − 12-s + 4·15-s − 3·16-s + 4·17-s − 4·20-s + 6·25-s − 27-s + 36-s + 4·37-s + 4·41-s − 8·43-s − 4·45-s − 16·47-s + 3·48-s − 4·51-s − 8·59-s + 4·60-s − 7·64-s − 8·67-s + 4·68-s − 6·75-s + 12·80-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.78·5-s + 1/3·9-s − 0.288·12-s + 1.03·15-s − 3/4·16-s + 0.970·17-s − 0.894·20-s + 6/5·25-s − 0.192·27-s + 1/6·36-s + 0.657·37-s + 0.624·41-s − 1.21·43-s − 0.596·45-s − 2.33·47-s + 0.433·48-s − 0.560·51-s − 1.04·59-s + 0.516·60-s − 7/8·64-s − 0.977·67-s + 0.485·68-s − 0.692·75-s + 1.34·80-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.698223626536844325957221893577, −9.166413921251398402501648146432, −8.451250846047373159735122750477, −7.909404764979961692250342151734, −7.71727710509481961360736996674, −7.06949902462298002679091800838, −6.61885560247283937341370127897, −6.05280851814515950319512149656, −5.28963474878067664613973490473, −4.64913877666350581776645513241, −4.15612071791617709049738843477, −3.45591764213572954828675776871, −2.82787864773804551728997226277, −1.51807799863530483435123120604, 0,
1.51807799863530483435123120604, 2.82787864773804551728997226277, 3.45591764213572954828675776871, 4.15612071791617709049738843477, 4.64913877666350581776645513241, 5.28963474878067664613973490473, 6.05280851814515950319512149656, 6.61885560247283937341370127897, 7.06949902462298002679091800838, 7.71727710509481961360736996674, 7.909404764979961692250342151734, 8.451250846047373159735122750477, 9.166413921251398402501648146432, 9.698223626536844325957221893577
