| L(s) = 1 | + (0.684 + 1.59i)3-s + (1.15 + 3.17i)5-s + (−0.593 − 3.36i)7-s + (−2.06 + 2.17i)9-s + (−0.197 + 0.541i)11-s + (−4.30 + 5.12i)13-s + (−4.25 + 4.00i)15-s + (1.15 − 2.00i)17-s + (−0.353 + 0.203i)19-s + (4.94 − 3.24i)21-s + (−1.15 + 6.53i)23-s + (−4.89 + 4.10i)25-s + (−4.87 − 1.78i)27-s + (2.24 + 2.67i)29-s + (0.382 − 2.17i)31-s + ⋯ |
| L(s) = 1 | + (0.395 + 0.918i)3-s + (0.516 + 1.41i)5-s + (−0.224 − 1.27i)7-s + (−0.687 + 0.726i)9-s + (−0.0594 + 0.163i)11-s + (−1.19 + 1.42i)13-s + (−1.09 + 1.03i)15-s + (0.281 − 0.486i)17-s + (−0.0810 + 0.0467i)19-s + (1.07 − 0.708i)21-s + (−0.240 + 1.36i)23-s + (−0.978 + 0.821i)25-s + (−0.938 − 0.344i)27-s + (0.417 + 0.497i)29-s + (0.0687 − 0.390i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.501257 + 1.40543i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.501257 + 1.40543i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.684 - 1.59i)T \) |
| good | 5 | \( 1 + (-1.15 - 3.17i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.593 + 3.36i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.197 - 0.541i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (4.30 - 5.12i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.15 + 2.00i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.353 - 0.203i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.15 - 6.53i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.24 - 2.67i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.382 + 2.17i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.05 - 0.607i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.09 - 4.27i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.442 + 1.21i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.547 - 3.10i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 9.35iT - 53T^{2} \) |
| 59 | \( 1 + (3.00 + 8.25i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-4.67 + 0.824i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (9.67 - 11.5i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.92 + 6.79i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.641 - 1.11i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.84 - 3.22i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.67 - 7.95i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.86 + 4.96i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.5 - 3.84i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.21264447401798049669210916410, −9.869642480006733790355502572542, −9.210611057580536176564930660666, −7.68972082224715119717828880580, −7.16908960886273509397487216476, −6.31808508747056081143195639746, −5.01344344579863522550812942529, −4.05974079195618533960732597883, −3.17928562297354928361776007600, −2.14188617283269461246576231098,
0.65732007729384735448578499109, 2.12230111163878232212112996151, 2.92063780519692672621087484192, 4.66399915646043293880705059654, 5.64828240368355645213355896975, 6.07585235654954416917401351236, 7.48477424347453827903319194529, 8.335323038044559186240430923141, 8.803933306917330165861183171576, 9.568880056494365607596355358589
