| L(s) = 1 | + (25.8 + 49.5i)5-s + 154.·7-s + 104.·11-s + 727. i·13-s + 484.·17-s + 877. i·19-s + 1.02e3i·23-s + (−1.78e3 + 2.56e3i)25-s + 6.37e3i·29-s − 7.63e3i·31-s + (4.00e3 + 7.67e3i)35-s + 5.54e3i·37-s − 4.74e3i·41-s − 4.32e3·43-s + 1.62e4i·47-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.886i)5-s + 1.19·7-s + 0.259·11-s + 1.19i·13-s + 0.406·17-s + 0.557i·19-s + 0.404i·23-s + (−0.571 + 0.820i)25-s + 1.40i·29-s − 1.42i·31-s + (0.553 + 1.05i)35-s + 0.666i·37-s − 0.440i·41-s − 0.357·43-s + 1.07i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.659974176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.659974176\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-25.8 - 49.5i)T \) |
| good | 7 | \( 1 - 154.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 104.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 727. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 484.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 877. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.02e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 6.37e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.63e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 5.54e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 4.74e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 4.32e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.62e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.18e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.01e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.25e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.70e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 9.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.41e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.24e5iT - 8.58e9T^{2} \) |
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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
−9.905190403380359303119077878318, −9.153204940425013838414914294179, −8.104963006038570238336341154875, −7.30061967343110205618636261178, −6.44101510659407872851863056435, −5.49697801101376124212857286585, −4.46965437835555811962045591684, −3.41013045853274778401782637586, −2.10530853271871728982738225407, −1.38156910854272029860715354340,
0.54582766717048438309778450237, 1.43261307306698712357843783044, 2.54650000098733021263298904908, 3.99429567378212386378450842039, 5.03980527721180105990086575346, 5.50362294790523100492203175478, 6.72148792682036245532283884615, 8.004805340440838348663106510497, 8.331189040535399099660548399197, 9.349272420198239954082359045232
