| L(s) = 1 | + (0.955 + 0.294i)2-s + (0.427 − 1.08i)3-s + (0.826 + 0.563i)4-s + (−0.0821 + 0.0123i)5-s + (0.729 − 0.914i)6-s + (−2.45 − 0.985i)7-s + (0.623 + 0.781i)8-s + (1.19 + 1.11i)9-s + (−0.0821 − 0.0123i)10-s + (−2.81 + 2.61i)11-s + (0.966 − 0.658i)12-s + (−0.384 − 1.68i)13-s + (−2.05 − 1.66i)14-s + (−0.0216 + 0.0947i)15-s + (0.365 + 0.930i)16-s + (−0.107 − 1.43i)17-s + ⋯ |
| L(s) = 1 | + (0.675 + 0.208i)2-s + (0.246 − 0.628i)3-s + (0.413 + 0.281i)4-s + (−0.0367 + 0.00553i)5-s + (0.297 − 0.373i)6-s + (−0.927 − 0.372i)7-s + (0.220 + 0.276i)8-s + (0.398 + 0.370i)9-s + (−0.0259 − 0.00391i)10-s + (−0.848 + 0.787i)11-s + (0.278 − 0.190i)12-s + (−0.106 − 0.466i)13-s + (−0.549 − 0.445i)14-s + (−0.00558 + 0.0244i)15-s + (0.0913 + 0.232i)16-s + (−0.0260 − 0.347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39629 - 0.0856835i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39629 - 0.0856835i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.955 - 0.294i)T \) |
| 7 | \( 1 + (2.45 + 0.985i)T \) |
| good | 3 | \( 1 + (-0.427 + 1.08i)T + (-2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (0.0821 - 0.0123i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (2.81 - 2.61i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.384 + 1.68i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.107 + 1.43i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (1.48 - 2.56i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.117 + 1.56i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-4.85 - 2.34i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (5.30 + 9.17i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.91 + 4.71i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (-2.53 - 3.17i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (0.652 - 0.817i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-8.15 - 2.51i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (-2.84 - 1.94i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-3.45 - 0.520i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (-0.521 + 0.355i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (5.32 + 9.22i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.00 - 1.92i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (7.73 - 2.38i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-6.16 + 10.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.19 - 14.0i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (11.3 + 10.5i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−13.62070668136370569661320686432, −12.97264048596067698950575266563, −12.32395091362016640691284055782, −10.72733659262635329362157253469, −9.704495863782537532758157530495, −7.86935982901573571531051378204, −7.19620649920694086874670116794, −5.85897507409912874451224769259, −4.26530578057937287608860883437, −2.51812298702047984220304781566,
2.90312413154170727052021560636, 4.15438312330091793667759772115, 5.66757634375490141555499370060, 6.89550790419969518139018305746, 8.657879737435610526177165398000, 9.790098426522978838188075518924, 10.69651774992616681182487002987, 12.00511878678345490599069790594, 12.96314256101413795723799509573, 13.84593600730948078050820650442
