| L(s) = 1 | − 3-s − 2·5-s − 2·11-s + 6·13-s + 2·15-s + 8·17-s − 19-s − 8·23-s + 5·25-s + 27-s + 8·29-s + 3·31-s + 2·33-s + 37-s − 6·39-s − 12·41-s + 22·43-s + 6·47-s − 8·51-s + 12·53-s + 4·55-s + 57-s + 4·59-s − 6·61-s − 12·65-s − 13·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.603·11-s + 1.66·13-s + 0.516·15-s + 1.94·17-s − 0.229·19-s − 1.66·23-s + 25-s + 0.192·27-s + 1.48·29-s + 0.538·31-s + 0.348·33-s + 0.164·37-s − 0.960·39-s − 1.87·41-s + 3.35·43-s + 0.875·47-s − 1.12·51-s + 1.64·53-s + 0.539·55-s + 0.132·57-s + 0.520·59-s − 0.768·61-s − 1.48·65-s − 1.58·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.263153378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263153378\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.67474349052871886641334513973, −10.46581362455315820703270200404, −10.38655405434576491905132926398, −9.774325998618023450521222488917, −8.895836238776167463077657651039, −8.793829749877786594326541294749, −8.078314047019151047749279353773, −7.999192546505736906353541738593, −7.35687613609293183882966911755, −6.99022940396525788328374918806, −6.17381983040674805170905629738, −5.83681122472412503926110971329, −5.66878862298854949462179269936, −4.83450555496741839915063303491, −4.15554911104864310336645817503, −3.98277885183252021178572561827, −3.11440067935756924438277152789, −2.73572212186900775081370168887, −1.46196899898935132217137690139, −0.72500536789799164666846476553,
0.72500536789799164666846476553, 1.46196899898935132217137690139, 2.73572212186900775081370168887, 3.11440067935756924438277152789, 3.98277885183252021178572561827, 4.15554911104864310336645817503, 4.83450555496741839915063303491, 5.66878862298854949462179269936, 5.83681122472412503926110971329, 6.17381983040674805170905629738, 6.99022940396525788328374918806, 7.35687613609293183882966911755, 7.999192546505736906353541738593, 8.078314047019151047749279353773, 8.793829749877786594326541294749, 8.895836238776167463077657651039, 9.774325998618023450521222488917, 10.38655405434576491905132926398, 10.46581362455315820703270200404, 10.67474349052871886641334513973
