| L(s) = 1 | + (−0.173 − 0.984i)2-s + (1.93 − 1.61i)3-s + (−0.939 + 0.342i)4-s + (2.31 + 0.841i)5-s + (−1.93 − 1.61i)6-s + (1.39 − 2.41i)7-s + (0.5 + 0.866i)8-s + (0.581 − 3.30i)9-s + (0.427 − 2.42i)10-s + (0.839 + 1.45i)11-s + (−1.26 + 2.18i)12-s + (−4.86 − 4.07i)13-s + (−2.62 − 0.955i)14-s + (5.82 − 2.12i)15-s + (0.766 − 0.642i)16-s + (0.862 + 4.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (1.11 − 0.935i)3-s + (−0.469 + 0.171i)4-s + (1.03 + 0.376i)5-s + (−0.788 − 0.661i)6-s + (0.527 − 0.914i)7-s + (0.176 + 0.306i)8-s + (0.193 − 1.10i)9-s + (0.135 − 0.765i)10-s + (0.253 + 0.438i)11-s + (−0.363 + 0.630i)12-s + (−1.34 − 1.13i)13-s + (−0.701 − 0.255i)14-s + (1.50 − 0.547i)15-s + (0.191 − 0.160i)16-s + (0.209 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39940 - 1.89912i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39940 - 1.89912i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-1.93 + 1.61i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-2.31 - 0.841i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.39 + 2.41i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.839 - 1.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.86 + 4.07i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.862 - 4.89i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.34 + 0.854i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.03 - 5.84i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.64 + 6.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.550T + 37T^{2} \) |
| 41 | \( 1 + (-1.99 + 1.67i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.69 + 0.982i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.129 + 0.734i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (1.38 - 0.502i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.861 - 4.88i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (8.77 - 3.19i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.00 - 11.3i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (6.57 + 2.39i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (4.73 - 3.97i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.52 + 3.79i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.58 - 13.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.28 + 4.43i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.49 + 14.1i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.22114727843666696089829602483, −9.396402141266566754191476374693, −8.387483051608356278718227345471, −7.62503388574973216818244000699, −6.99739094057406455677311651054, −5.68745446801773948711613682752, −4.39650858350823070388954092638, −3.08193842932873273527756691611, −2.23737334952519528636303436698, −1.30318593753012164420155021520,
1.98129561244858413131812119229, 3.02145420597799208369515910117, 4.58920378692015898487899137866, 5.06654778384422246794605268681, 6.14943430841513559534243997844, 7.32426826662340250193432579726, 8.362253973237869014802069501767, 9.128187946042703674664930858696, 9.443041973948365585101047655857, 10.07675676715023038428416436023
