| L(s) = 1 | + (−1.39 − 0.212i)2-s + (1.33 − 1.10i)3-s + (1.90 + 0.595i)4-s + (−0.168 + 0.574i)5-s + (−2.10 + 1.25i)6-s + (−2.73 − 2.36i)7-s + (−2.54 − 1.23i)8-s + (0.567 − 2.94i)9-s + (0.358 − 0.767i)10-s + (2.81 − 1.81i)11-s + (3.20 − 1.31i)12-s + (−2.47 − 2.85i)13-s + (3.31 + 3.89i)14-s + (0.408 + 0.953i)15-s + (3.29 + 2.27i)16-s + (1.55 + 0.708i)17-s + ⋯ |
| L(s) = 1 | + (−0.988 − 0.150i)2-s + (0.771 − 0.636i)3-s + (0.954 + 0.297i)4-s + (−0.0754 + 0.257i)5-s + (−0.858 + 0.513i)6-s + (−1.03 − 0.894i)7-s + (−0.898 − 0.437i)8-s + (0.189 − 0.981i)9-s + (0.113 − 0.242i)10-s + (0.849 − 0.546i)11-s + (0.925 − 0.378i)12-s + (−0.685 − 0.790i)13-s + (0.886 + 1.04i)14-s + (0.105 + 0.246i)15-s + (0.822 + 0.568i)16-s + (0.376 + 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.616184 - 0.686845i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.616184 - 0.686845i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.212i)T \) |
| 3 | \( 1 + (-1.33 + 1.10i)T \) |
| 23 | \( 1 + (-3.51 + 3.26i)T \) |
| good | 5 | \( 1 + (0.168 - 0.574i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (2.73 + 2.36i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-2.81 + 1.81i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.47 + 2.85i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.55 - 0.708i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.386 - 0.176i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.0545 - 0.0249i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (7.54 + 1.08i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-9.04 + 2.65i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.14 - 7.29i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-3.74 + 0.537i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 8.85T + 47T^{2} \) |
| 53 | \( 1 + (-2.10 - 1.82i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.59 - 4.14i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.40 - 9.79i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (4.30 - 6.69i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-11.2 - 7.21i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.421 + 0.923i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-9.95 + 8.63i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (8.26 - 2.42i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-14.9 + 2.14i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.21 - 1.82i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.50756409214049258221840214039, −10.46700159349001616742895051591, −9.613853107063599078093493739706, −8.837561898895485484371418967569, −7.70114831315481128839820770657, −7.01780821923986556455656502696, −6.18477154034029614668637731755, −3.66642759270570567527825072641, −2.80804472106966326847018355623, −0.907537125508929923071057605361,
2.12781495904764288001414114601, 3.38954758682261550969418292820, 5.03791879317156148861658062244, 6.46260902474772263508670760010, 7.42559419050421171777248839469, 8.650081397318038372767199277108, 9.502299326503151509228352798507, 9.575777706550128542963486473307, 10.97048063688467511439551228061, 12.03538160575771150877664773523
