| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 4·5-s + 9·6-s − 3·7-s − 3·8-s + 4·9-s + 12·10-s − 5·11-s − 12·12-s − 5·13-s + 9·14-s + 12·15-s + 3·16-s − 2·17-s − 12·18-s − 19-s − 16·20-s + 9·21-s + 15·22-s + 2·23-s + 9·24-s + 4·25-s + 15·26-s − 6·27-s − 12·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 1.78·5-s + 3.67·6-s − 1.13·7-s − 1.06·8-s + 4/3·9-s + 3.79·10-s − 1.50·11-s − 3.46·12-s − 1.38·13-s + 2.40·14-s + 3.09·15-s + 3/4·16-s − 0.485·17-s − 2.82·18-s − 0.229·19-s − 3.57·20-s + 1.96·21-s + 3.19·22-s + 0.417·23-s + 1.83·24-s + 4/5·25-s + 2.94·26-s − 1.15·27-s − 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5599 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5599 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 509 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 30 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 57 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 55 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T - 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 80 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 83 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T - 8 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 17 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−17.6785531667, −17.4653460943, −16.9088551486, −16.5933846413, −16.1318738086, −15.6009204218, −15.3398556905, −14.7602980808, −13.3215146281, −13.0130173839, −12.1846433284, −11.9543233778, −11.3585972689, −10.8763934076, −10.3429052558, −9.75382883150, −9.46889586787, −8.41912346706, −7.89102471063, −7.55101399568, −6.84125100639, −6.06885033074, −5.17499439430, −4.34806337685, −2.97400679081, 0, 0,
2.97400679081, 4.34806337685, 5.17499439430, 6.06885033074, 6.84125100639, 7.55101399568, 7.89102471063, 8.41912346706, 9.46889586787, 9.75382883150, 10.3429052558, 10.8763934076, 11.3585972689, 11.9543233778, 12.1846433284, 13.0130173839, 13.3215146281, 14.7602980808, 15.3398556905, 15.6009204218, 16.1318738086, 16.5933846413, 16.9088551486, 17.4653460943, 17.6785531667
