| L(s) = 1 | + 2-s + 2·5-s − 7-s + 8-s + 2·10-s + 11-s − 3·13-s − 14-s − 16-s + 17-s − 3·19-s + 22-s + 2·25-s − 3·26-s + 5·29-s − 4·31-s − 6·32-s + 34-s − 2·35-s − 8·37-s − 3·38-s + 2·40-s − 41-s + 9·43-s + 47-s + 5·49-s + 2·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s − 0.832·13-s − 0.267·14-s − 1/4·16-s + 0.242·17-s − 0.688·19-s + 0.213·22-s + 2/5·25-s − 0.588·26-s + 0.928·29-s − 0.718·31-s − 1.06·32-s + 0.171·34-s − 0.338·35-s − 1.31·37-s − 0.486·38-s + 0.316·40-s − 0.156·41-s + 1.37·43-s + 0.145·47-s + 5/7·49-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10935 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10935 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.483498365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483498365\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 88 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 116 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 157 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−16.5511501673, −15.9051882331, −15.4889455021, −14.8735698922, −14.2433565453, −14.0555388100, −13.6559092155, −12.9113374422, −12.7065402618, −12.1499418942, −11.5571430455, −10.6730675528, −10.4966435668, −9.78325865756, −9.21708769501, −8.79847218642, −7.94425674008, −7.10349564003, −6.76909330096, −5.89204611923, −5.38511803914, −4.62185171166, −3.99702297912, −2.90019871338, −1.91783444976,
1.91783444976, 2.90019871338, 3.99702297912, 4.62185171166, 5.38511803914, 5.89204611923, 6.76909330096, 7.10349564003, 7.94425674008, 8.79847218642, 9.21708769501, 9.78325865756, 10.4966435668, 10.6730675528, 11.5571430455, 12.1499418942, 12.7065402618, 12.9113374422, 13.6559092155, 14.0555388100, 14.2433565453, 14.8735698922, 15.4889455021, 15.9051882331, 16.5511501673
