| L(s) = 1 | + (−1.58 + 0.336i)2-s + (−0.104 − 0.994i)3-s + (0.564 − 0.251i)4-s + (1.49 − 1.66i)5-s + (0.5 + 1.53i)6-s + (−1.51 − 2.17i)7-s + (1.80 − 1.31i)8-s + (1.95 − 0.415i)9-s + (−1.80 + 3.13i)10-s + (0.988 − 3.16i)11-s + (−0.309 − 0.535i)12-s + (−1.80 + 5.56i)13-s + (3.12 + 2.92i)14-s + (−1.80 − 1.31i)15-s + (−3.24 + 3.60i)16-s + (−3.16 − 0.672i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 + 0.237i)2-s + (−0.0603 − 0.574i)3-s + (0.282 − 0.125i)4-s + (0.669 − 0.743i)5-s + (0.204 + 0.628i)6-s + (−0.572 − 0.820i)7-s + (0.639 − 0.464i)8-s + (0.652 − 0.138i)9-s + (−0.572 + 0.990i)10-s + (0.298 − 0.954i)11-s + (−0.0892 − 0.154i)12-s + (−0.501 + 1.54i)13-s + (0.835 + 0.781i)14-s + (−0.467 − 0.339i)15-s + (−0.812 + 0.901i)16-s + (−0.767 − 0.163i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.498503 - 0.262064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.498503 - 0.262064i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.51 + 2.17i)T \) |
| 11 | \( 1 + (-0.988 + 3.16i)T \) |
| good | 2 | \( 1 + (1.58 - 0.336i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (0.104 + 0.994i)T + (-2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.49 + 1.66i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (1.80 - 5.56i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (3.16 + 0.672i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-1.04 - 0.466i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.881 - 1.52i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.35 - 4.61i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.93 - 3.25i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.636 + 6.05i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-1.92 + 1.40i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.145T + 43T^{2} \) |
| 47 | \( 1 + (-0.431 - 0.192i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.07 - 4.52i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (8.86 - 3.94i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (7.32 - 8.13i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (4.19 - 7.25i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.236 - 0.726i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-9.91 + 4.41i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-12.4 + 2.64i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (2.78 + 8.55i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.763 + 1.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.19 - 3.66i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−13.88062671767744405652814386472, −13.42205916250291889345837923987, −12.25144517691020986914468597954, −10.66453557760668184811073259074, −9.482297533417378240501132364228, −8.882436616308005657364985489603, −7.33397581815041617496586385728, −6.51236454410943752405281615498, −4.36148559571460170562599272434, −1.26071485104049961258309706927,
2.48967034069081766862086709704, 4.87002755717468373879739907980, 6.55355054841958374355120279511, 8.025037855768917168022721057430, 9.517467583382559347491885339115, 9.962723910405268393255808171170, 10.76899772091727524068118366014, 12.36147315523575167117225354469, 13.55475107883603655615679209311, 15.02294631891645279977990106467
