L(s) = 1 | + (−0.0556 − 0.0697i)3-s + (−3.27 + 2.61i)5-s + (1.51 − 2.17i)7-s + (0.665 − 2.91i)9-s + (−1.29 + 0.296i)11-s + (1.55 − 0.354i)13-s + (0.364 + 0.0832i)15-s + (−0.356 + 0.739i)17-s + 5.85·19-s + (−0.235 + 0.0152i)21-s + (1.52 + 3.16i)23-s + (2.79 − 12.2i)25-s + (−0.481 + 0.232i)27-s + (0.642 + 0.309i)29-s + 9.57·31-s + ⋯ |
L(s) = 1 | + (−0.0321 − 0.0402i)3-s + (−1.46 + 1.16i)5-s + (0.571 − 0.820i)7-s + (0.221 − 0.972i)9-s + (−0.391 + 0.0893i)11-s + (0.431 − 0.0984i)13-s + (0.0941 + 0.0214i)15-s + (−0.0863 + 0.179i)17-s + 1.34·19-s + (−0.0514 + 0.00332i)21-s + (0.317 + 0.659i)23-s + (0.558 − 2.44i)25-s + (−0.0927 + 0.0446i)27-s + (0.119 + 0.0574i)29-s + 1.71·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24286 - 0.126074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24286 - 0.126074i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.51 + 2.17i)T \) |
good | 3 | \( 1 + (0.0556 + 0.0697i)T + (-0.667 + 2.92i)T^{2} \) |
| 5 | \( 1 + (3.27 - 2.61i)T + (1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (1.29 - 0.296i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-1.55 + 0.354i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (0.356 - 0.739i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + (-1.52 - 3.16i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-0.642 - 0.309i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 - 9.57T + 31T^{2} \) |
| 37 | \( 1 + (4.49 + 2.16i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (-8.02 + 6.39i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-4.98 - 3.97i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-2.26 - 9.93i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-5.11 + 2.46i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (-2.50 + 3.13i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (4.17 - 8.66i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 12.7iT - 67T^{2} \) |
| 71 | \( 1 + (3.72 + 7.73i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-4.61 - 1.05i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 - 3.67iT - 79T^{2} \) |
| 83 | \( 1 + (-2.48 + 10.8i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (13.2 + 3.03i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 - 9.96iT - 97T^{2} \) |
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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
−10.53129865146485423821470428352, −9.520153604462140269098168392081, −8.296421578720990625449211708333, −7.50274020508243192951929337450, −7.11072331283599040766982518733, −6.05140535920930185324864422821, −4.55453789874824219121094169944, −3.74203692040975255233033297667, −2.97720694564552839244076845529, −0.864817037722735177472302757967,
1.07216411798247218293459799045, 2.72069535100091066541781299037, 4.13952263923913081160324411916, 4.90348589525738111005084640807, 5.53250080208048181139335126210, 7.14238008545240829222851191781, 8.035852271240300846946130917205, 8.385522569942478510777810090947, 9.228413365473156283996156192281, 10.43845248867276104082657419900