| L(s) = 1 | + 2-s − 2·5-s + 7-s − 8-s − 2·10-s − 4·11-s − 6·13-s + 14-s − 16-s − 4·17-s − 8·19-s − 4·22-s + 8·23-s + 5·25-s − 6·26-s − 2·29-s − 4·34-s − 2·35-s − 20·37-s − 8·38-s + 2·40-s − 6·41-s + 4·43-s + 8·46-s + 5·50-s − 12·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s − 1.20·11-s − 1.66·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s − 0.852·22-s + 1.66·23-s + 25-s − 1.17·26-s − 0.371·29-s − 0.685·34-s − 0.338·35-s − 3.28·37-s − 1.29·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s + 1.17·46-s + 0.707·50-s − 1.64·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2763208085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2763208085\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−10.49490286720802973267046654617, −9.414940994046001405079471365323, −9.225156556763939278066647661870, −8.589052375645920029969326396141, −8.560338148998537164405704964220, −7.76977622800345579708962766913, −7.68800240946826680593727913074, −6.96370913749017170345196380992, −6.76432306362789556021490776602, −6.42619518965534409040733330419, −5.42631066592115025936896753413, −5.14018480024727090237070710375, −4.97060345755627881930025798668, −4.39605801767808140083202818606, −4.08321840651976725638466863125, −3.25438324421925786745887771407, −2.94731104322690226353970298812, −2.28836752178602409335801169070, −1.73703919101765279046127405963, −0.19229586220280067418553044688,
0.19229586220280067418553044688, 1.73703919101765279046127405963, 2.28836752178602409335801169070, 2.94731104322690226353970298812, 3.25438324421925786745887771407, 4.08321840651976725638466863125, 4.39605801767808140083202818606, 4.97060345755627881930025798668, 5.14018480024727090237070710375, 5.42631066592115025936896753413, 6.42619518965534409040733330419, 6.76432306362789556021490776602, 6.96370913749017170345196380992, 7.68800240946826680593727913074, 7.76977622800345579708962766913, 8.560338148998537164405704964220, 8.589052375645920029969326396141, 9.225156556763939278066647661870, 9.414940994046001405079471365323, 10.49490286720802973267046654617
