| L(s) = 1 | − 3·5-s − 14·7-s + 15·11-s − 51·17-s + 27·19-s − 9·23-s − 19·25-s + 12·29-s − 21·31-s + 42·35-s − 31·37-s + 20·43-s − 75·47-s + 147·49-s − 57·53-s − 45·55-s + 141·59-s − 141·61-s + 49·67-s + 252·71-s − 45·73-s − 210·77-s + 73·79-s + 153·85-s − 99·89-s − 81·95-s − 171·101-s + ⋯ |
| L(s) = 1 | − 3/5·5-s − 2·7-s + 1.36·11-s − 3·17-s + 1.42·19-s − 0.391·23-s − 0.759·25-s + 0.413·29-s − 0.677·31-s + 6/5·35-s − 0.837·37-s + 0.465·43-s − 1.59·47-s + 3·49-s − 1.07·53-s − 0.818·55-s + 2.38·59-s − 2.31·61-s + 0.731·67-s + 3.54·71-s − 0.616·73-s − 2.72·77-s + 0.924·79-s + 9/5·85-s − 1.11·89-s − 0.852·95-s − 1.69·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7192850993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7192850993\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 28 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T + 104 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 27 T + 604 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T - 448 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 21 T + 1108 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 31 T - 408 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 75 T + 4084 T^{2} + 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 57 T + 440 T^{2} + 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 141 T + 10108 T^{2} - 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 141 T + 10348 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 49 T - 2088 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 126 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 45 T + 6004 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 73 T - 912 T^{2} - 73 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13586 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 99 T + 11188 T^{2} + 99 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18050 T^{2} + p^{4} T^{4} \) |
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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
−12.14032817765869011957983604958, −11.41861929513487665162982187485, −11.32549006934264432069563201644, −10.71763833675885394970928104830, −9.973832237965781128276536914146, −9.463792613907967534061815253397, −9.389145471249842429347125579912, −8.770534289910427412457278854722, −8.294534183359451922712600024594, −7.45657614734731138842171110743, −6.92346821483163259799608941517, −6.50118779480386688174716008734, −6.35679581020624524347951091886, −5.45582835975907456207912509093, −4.64758161690915555562038815346, −3.84024627332773888221688236604, −3.70877078439006469286359745513, −2.83144533122841435129863495280, −1.92190421378472824187386611630, −0.43038815973417211266340839360,
0.43038815973417211266340839360, 1.92190421378472824187386611630, 2.83144533122841435129863495280, 3.70877078439006469286359745513, 3.84024627332773888221688236604, 4.64758161690915555562038815346, 5.45582835975907456207912509093, 6.35679581020624524347951091886, 6.50118779480386688174716008734, 6.92346821483163259799608941517, 7.45657614734731138842171110743, 8.294534183359451922712600024594, 8.770534289910427412457278854722, 9.389145471249842429347125579912, 9.463792613907967534061815253397, 9.973832237965781128276536914146, 10.71763833675885394970928104830, 11.32549006934264432069563201644, 11.41861929513487665162982187485, 12.14032817765869011957983604958
