| L(s) = 1 | + (−3.91 − 1.15i)5-s + (−0.0825 + 0.0952i)7-s + (1.03 − 0.667i)11-s + (1.25 + 1.44i)13-s + (−0.787 + 1.72i)17-s + (0.0613 + 0.134i)19-s + (2.15 + 4.28i)23-s + (9.83 + 6.31i)25-s + (−3.11 + 6.82i)29-s + (0.335 − 2.33i)31-s + (0.433 − 0.278i)35-s + (−7.41 + 2.17i)37-s + (10.0 + 2.95i)41-s + (1.71 + 11.9i)43-s + 5.71·47-s + ⋯ |
| L(s) = 1 | + (−1.75 − 0.514i)5-s + (−0.0311 + 0.0359i)7-s + (0.313 − 0.201i)11-s + (0.347 + 0.400i)13-s + (−0.190 + 0.418i)17-s + (0.0140 + 0.0308i)19-s + (0.448 + 0.893i)23-s + (1.96 + 1.26i)25-s + (−0.578 + 1.26i)29-s + (0.0602 − 0.419i)31-s + (0.0731 − 0.0470i)35-s + (−1.21 + 0.357i)37-s + (1.56 + 0.460i)41-s + (0.261 + 1.82i)43-s + 0.833·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.679857 + 0.475025i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679857 + 0.475025i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-2.15 - 4.28i)T \) |
| good | 5 | \( 1 + (3.91 + 1.15i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.0825 - 0.0952i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.03 + 0.667i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 1.44i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.787 - 1.72i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.0613 - 0.134i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.11 - 6.82i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.335 + 2.33i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (7.41 - 2.17i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-10.0 - 2.95i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.71 - 11.9i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 5.71T + 47T^{2} \) |
| 53 | \( 1 + (-4.37 + 5.04i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.98 - 5.75i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.180 - 1.25i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (8.94 + 5.75i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (7.77 + 4.99i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.25 + 11.4i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-2.44 - 2.81i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (7.82 - 2.29i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.00 + 6.98i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-12.7 - 3.73i)T + (81.6 + 52.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.64423020516331146943322662258, −9.208085887226485834817654108279, −8.750789834507264666679041279087, −7.76583704563910906482250793195, −7.21867275202786495630529322751, −6.03438874053048803199948363283, −4.80790415266157306977496919841, −4.01301543923235881196290027394, −3.19659360244088550739369449124, −1.23070716648933600391145194644,
0.47596432696421902638485467390, 2.60482380445250460518891485604, 3.74589553158048160915146806186, 4.33835628382551518342487099776, 5.64393543070951507227963522692, 6.96139066226343287179313254386, 7.34206657280805778166452034855, 8.350062423400882739839516626413, 8.980720720000046912085697668748, 10.32835315802982911516502574670
