| L(s) = 1 | + 3·2-s − 6·3-s + 4-s + 10·5-s − 18·6-s + 3·8-s + 27·9-s + 30·10-s + 62·11-s − 6·12-s + 6·13-s − 60·15-s + 9·16-s − 40·17-s + 81·18-s + 122·19-s + 10·20-s + 186·22-s + 16·23-s − 18·24-s + 75·25-s + 18·26-s − 108·27-s + 352·29-s − 180·30-s − 66·31-s − 165·32-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 1.15·3-s + 1/8·4-s + 0.894·5-s − 1.22·6-s + 0.132·8-s + 9-s + 0.948·10-s + 1.69·11-s − 0.144·12-s + 0.128·13-s − 1.03·15-s + 9/64·16-s − 0.570·17-s + 1.06·18-s + 1.47·19-s + 0.111·20-s + 1.80·22-s + 0.145·23-s − 0.153·24-s + 3/5·25-s + 0.135·26-s − 0.769·27-s + 2.25·29-s − 1.09·30-s − 0.382·31-s − 0.911·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.536928785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.536928785\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + p^{3} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 62 T + 3582 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 3378 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 40 T + 8750 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 122 T + 17398 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 16 T + 34 p T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 352 T + 78278 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 66 T + 45870 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 188 T + 44542 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 18350 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 396 T + 2226 p T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 p T + 15582 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 982 T + 504354 T^{2} - 982 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 516 T + 418118 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 880 T + 575238 T^{2} - 880 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 356 T + 100374 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 310 T + 664038 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 326 T + 789802 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1832 T + 1824478 T^{2} - 1832 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 680 T + 744870 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 796 T + 411158 T^{2} + 796 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 670 T + 1145410 T^{2} - 670 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.26243306453489764673215649701, −9.958879774976066977732556786045, −9.312958750931259854264478827614, −9.091875961957082198469817865672, −8.619662345738057846624240375741, −8.014715513445701045886512751649, −7.23424147500228175669407005459, −6.80558046091901650361741660254, −6.69285609841721164681306558959, −6.15139409839427904295959889246, −5.53996684522321551081043933933, −5.33976194384292118421124142230, −4.74045759646112783460902802576, −4.51515373973666636588781518725, −3.73929958869782030011687353693, −3.52199884634867288140851344149, −2.55989918564782867568502447585, −1.79147931048963323611244451144, −1.17920045757108547304782338798, −0.67434224067588292521990695935,
0.67434224067588292521990695935, 1.17920045757108547304782338798, 1.79147931048963323611244451144, 2.55989918564782867568502447585, 3.52199884634867288140851344149, 3.73929958869782030011687353693, 4.51515373973666636588781518725, 4.74045759646112783460902802576, 5.33976194384292118421124142230, 5.53996684522321551081043933933, 6.15139409839427904295959889246, 6.69285609841721164681306558959, 6.80558046091901650361741660254, 7.23424147500228175669407005459, 8.014715513445701045886512751649, 8.619662345738057846624240375741, 9.091875961957082198469817865672, 9.312958750931259854264478827614, 9.958879774976066977732556786045, 10.26243306453489764673215649701
