| L(s) = 1 | + (0.777 + 0.629i)2-s + (0.393 + 1.68i)3-s + (0.207 + 0.978i)4-s + (2.07 + 2.30i)5-s + (−0.755 + 1.55i)6-s + (1.05 + 0.404i)7-s + (−0.453 + 0.891i)8-s + (−2.68 + 1.32i)9-s + (0.162 + 3.09i)10-s + (−4.01 − 4.01i)11-s + (−1.56 + 0.735i)12-s + (2.83 − 4.91i)13-s + (0.565 + 0.978i)14-s + (−3.06 + 4.40i)15-s + (−0.913 + 0.406i)16-s + (1.90 + 1.23i)17-s + ⋯ |
| L(s) = 1 | + (0.549 + 0.444i)2-s + (0.227 + 0.973i)3-s + (0.103 + 0.489i)4-s + (0.927 + 1.03i)5-s + (−0.308 + 0.636i)6-s + (0.398 + 0.153i)7-s + (−0.160 + 0.315i)8-s + (−0.896 + 0.442i)9-s + (0.0513 + 0.978i)10-s + (−1.20 − 1.20i)11-s + (−0.452 + 0.212i)12-s + (0.787 − 1.36i)13-s + (0.151 + 0.261i)14-s + (−0.792 + 1.13i)15-s + (−0.228 + 0.101i)16-s + (0.462 + 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23517 + 1.75588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23517 + 1.75588i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.777 - 0.629i)T \) |
| 3 | \( 1 + (-0.393 - 1.68i)T \) |
| 61 | \( 1 + (-7.57 - 1.89i)T \) |
| good | 5 | \( 1 + (-2.07 - 2.30i)T + (-0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (-1.05 - 0.404i)T + (5.20 + 4.68i)T^{2} \) |
| 11 | \( 1 + (4.01 + 4.01i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.83 + 4.91i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.90 - 1.23i)T + (6.91 + 15.5i)T^{2} \) |
| 19 | \( 1 + (-2.24 + 5.03i)T + (-12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (2.27 + 4.45i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (2.37 - 8.87i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-6.43 - 7.94i)T + (-6.44 + 30.3i)T^{2} \) |
| 37 | \( 1 + (6.29 - 0.997i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (0.254 + 0.185i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.61 - 1.69i)T + (17.4 - 39.2i)T^{2} \) |
| 47 | \( 1 + (0.258 - 0.149i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.94 - 2.51i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-8.60 + 10.6i)T + (-12.2 - 57.7i)T^{2} \) |
| 67 | \( 1 + (6.32 + 0.331i)T + (66.6 + 7.00i)T^{2} \) |
| 71 | \( 1 + (0.571 - 0.0299i)T + (70.6 - 7.42i)T^{2} \) |
| 73 | \( 1 + (6.02 - 6.68i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (2.15 + 3.31i)T + (-32.1 + 72.1i)T^{2} \) |
| 83 | \( 1 + (-1.15 + 0.120i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (5.59 + 0.886i)T + (84.6 + 27.5i)T^{2} \) |
| 97 | \( 1 + (-0.588 - 0.0618i)T + (94.8 + 20.1i)T^{2} \) |
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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
−11.39305463044252019264874704578, −10.52932943058510086134694326604, −10.30424902560178471274610343753, −8.692302922376477661793548427124, −8.150089219072391362025824207945, −6.70272297416277667619380717206, −5.57845018241164416060003512668, −5.16503213876092989900175025765, −3.29977089602145191494792717853, −2.81555869761763599797364764715,
1.48437992304662596358286098904, 2.23736644502398150551772551011, 4.10493399877907342640976124989, 5.32794228611440232993673414658, 6.04606052650892164885633818880, 7.39037959232694484674018889111, 8.265413944036106319143017729694, 9.473861635052040054415139006270, 10.06464414779949289588172019244, 11.63073611991750472515810988892
