| L(s) = 1 | − 2·3-s − 4-s − 5-s + 3·7-s + 4·9-s − 3·11-s + 2·12-s + 2·13-s + 2·15-s − 3·16-s − 4·17-s − 19-s + 20-s − 6·21-s + 23-s − 25-s − 5·27-s − 3·28-s + 4·29-s + 12·31-s + 6·33-s − 3·35-s − 4·36-s − 5·37-s − 4·39-s − 18·41-s + 2·43-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.447·5-s + 1.13·7-s + 4/3·9-s − 0.904·11-s + 0.577·12-s + 0.554·13-s + 0.516·15-s − 3/4·16-s − 0.970·17-s − 0.229·19-s + 0.223·20-s − 1.30·21-s + 0.208·23-s − 1/5·25-s − 0.962·27-s − 0.566·28-s + 0.742·29-s + 2.15·31-s + 1.04·33-s − 0.507·35-s − 2/3·36-s − 0.821·37-s − 0.640·39-s − 2.81·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121089 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 181 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| 223 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 164 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 164 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 10 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 15 T + 131 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T - 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 167 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−14.0084636806, −13.4190814420, −13.2574477057, −12.9709625224, −12.0596405089, −11.8740394176, −11.5894273720, −11.0204369678, −10.7160335125, −10.1460143539, −9.96987449166, −9.11334417143, −8.49313269001, −8.30693335328, −7.85469145771, −7.02808513730, −6.68103900252, −6.28142190870, −5.39815189636, −4.87847764647, −4.70812409668, −4.18829141945, −3.29409268136, −2.24887898966, −1.33243076350, 0,
1.33243076350, 2.24887898966, 3.29409268136, 4.18829141945, 4.70812409668, 4.87847764647, 5.39815189636, 6.28142190870, 6.68103900252, 7.02808513730, 7.85469145771, 8.30693335328, 8.49313269001, 9.11334417143, 9.96987449166, 10.1460143539, 10.7160335125, 11.0204369678, 11.5894273720, 11.8740394176, 12.0596405089, 12.9709625224, 13.2574477057, 13.4190814420, 14.0084636806
