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
−10.32835315802982911516502574670, −8.980720720000046912085697668748, −8.350062423400882739839516626413, −7.34206657280805778166452034855, −6.96139066226343287179313254386, −5.64393543070951507227963522692, −4.33835628382551518342487099776, −3.74589553158048160915146806186, −2.60482380445250460518891485604, −0.47596432696421902638485467390,
1.23070716648933600391145194644, 3.19659360244088550739369449124, 4.01301543923235881196290027394, 4.80790415266157306977496919841, 6.03438874053048803199948363283, 7.21867275202786495630529322751, 7.76583704563910906482250793195, 8.750789834507264666679041279087, 9.208085887226485834817654108279, 10.64423020516331146943322662258