| L(s) = 1 | + 3-s − 2·4-s − 2·5-s − 2·7-s + 9-s + 2·11-s − 2·12-s − 4·13-s − 2·15-s + 4·16-s + 2·17-s + 4·20-s − 2·21-s − 10·23-s − 25-s + 27-s + 4·28-s − 3·31-s + 2·33-s + 4·35-s − 2·36-s − 5·37-s − 4·39-s − 6·41-s − 3·43-s − 4·44-s − 2·45-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.577·12-s − 1.10·13-s − 0.516·15-s + 16-s + 0.485·17-s + 0.894·20-s − 0.436·21-s − 2.08·23-s − 1/5·25-s + 0.192·27-s + 0.755·28-s − 0.538·31-s + 0.348·33-s + 0.676·35-s − 1/3·36-s − 0.821·37-s − 0.640·39-s − 0.937·41-s − 0.457·43-s − 0.603·44-s − 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 25 T + 288 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T - 110 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 202 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 133 T^{2} + p^{2} T^{4} \) |
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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
−15.4916935379, −14.7865298729, −14.4694272720, −14.1821280588, −13.6408465207, −13.0978695724, −12.6946467167, −12.1658788070, −11.8514902468, −11.4269766834, −10.2472039388, −10.1548018029, −9.73272390467, −9.09336453756, −8.66260031396, −8.05961091016, −7.64746660343, −7.14599522491, −6.34638781014, −5.72769801300, −4.87926093280, −4.31481908434, −3.60611880807, −3.28469828907, −1.92895251547, 0,
1.92895251547, 3.28469828907, 3.60611880807, 4.31481908434, 4.87926093280, 5.72769801300, 6.34638781014, 7.14599522491, 7.64746660343, 8.05961091016, 8.66260031396, 9.09336453756, 9.73272390467, 10.1548018029, 10.2472039388, 11.4269766834, 11.8514902468, 12.1658788070, 12.6946467167, 13.0978695724, 13.6408465207, 14.1821280588, 14.4694272720, 14.7865298729, 15.4916935379
