| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 2·9-s + 7·11-s + 4·12-s − 4·13-s − 4·16-s + 2·17-s + 4·18-s + 14·22-s + 2·23-s − 8·26-s + 6·27-s − 6·29-s − 3·31-s − 8·32-s + 14·33-s + 4·34-s + 4·36-s + 2·37-s − 8·39-s − 3·41-s + 14·44-s + 4·46-s + 4·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 2/3·9-s + 2.11·11-s + 1.15·12-s − 1.10·13-s − 16-s + 0.485·17-s + 0.942·18-s + 2.98·22-s + 0.417·23-s − 1.56·26-s + 1.15·27-s − 1.11·29-s − 0.538·31-s − 1.41·32-s + 2.43·33-s + 0.685·34-s + 2/3·36-s + 0.328·37-s − 1.28·39-s − 0.468·41-s + 2.11·44-s + 0.589·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.787238450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.787238450\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 27 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T - 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 107 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 13 T + 177 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−14.1950548921, −13.6887400834, −13.3582518774, −12.7468081463, −12.4432713446, −12.0539071829, −11.6007560498, −11.2461501003, −10.5459011808, −9.88926377921, −9.44620398289, −9.01521419617, −8.82186813657, −8.04206360193, −7.37860055912, −7.06643741783, −6.43676960345, −6.03147997198, −5.22159278653, −4.72914773976, −4.10638397707, −3.62058646971, −3.12874863244, −2.40380937890, −1.52959279120,
1.52959279120, 2.40380937890, 3.12874863244, 3.62058646971, 4.10638397707, 4.72914773976, 5.22159278653, 6.03147997198, 6.43676960345, 7.06643741783, 7.37860055912, 8.04206360193, 8.82186813657, 9.01521419617, 9.44620398289, 9.88926377921, 10.5459011808, 11.2461501003, 11.6007560498, 12.0539071829, 12.4432713446, 12.7468081463, 13.3582518774, 13.6887400834, 14.1950548921
