| L(s) = 1 | + (−0.424 + 1.34i)2-s + (−0.796 − 1.53i)3-s + (−1.63 − 1.14i)4-s + (2.61 − 2.14i)5-s + (2.41 − 0.421i)6-s + (−1.02 − 1.53i)7-s + (2.24 − 1.72i)8-s + (−1.73 + 2.44i)9-s + (1.78 + 4.44i)10-s + (−4.07 + 1.23i)11-s + (−0.455 + 3.43i)12-s + (−0.147 − 0.121i)13-s + (2.51 − 0.734i)14-s + (−5.39 − 2.31i)15-s + (1.37 + 3.75i)16-s + (−2.33 − 5.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.300 + 0.953i)2-s + (−0.459 − 0.888i)3-s + (−0.819 − 0.572i)4-s + (1.17 − 0.961i)5-s + (0.985 − 0.172i)6-s + (−0.388 − 0.581i)7-s + (0.792 − 0.610i)8-s + (−0.577 + 0.816i)9-s + (0.565 + 1.40i)10-s + (−1.22 + 0.372i)11-s + (−0.131 + 0.991i)12-s + (−0.0410 − 0.0336i)13-s + (0.671 − 0.196i)14-s + (−1.39 − 0.598i)15-s + (0.344 + 0.938i)16-s + (−0.566 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.440496 - 0.556698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.440496 - 0.556698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.424 - 1.34i)T \) |
| 3 | \( 1 + (0.796 + 1.53i)T \) |
| good | 5 | \( 1 + (-2.61 + 2.14i)T + (0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (1.02 + 1.53i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (4.07 - 1.23i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (0.147 + 0.121i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (2.33 + 5.63i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (7.96 - 0.784i)T + (18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-4.90 + 0.974i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-6.87 - 2.08i)T + (24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-1.59 + 1.59i)T - 31iT^{2} \) |
| 37 | \( 1 + (11.3 + 1.11i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-3.55 + 0.707i)T + (37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.907 + 1.69i)T + (-23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (2.29 + 5.53i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.969 + 0.294i)T + (44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-4.29 + 3.52i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (3.08 + 5.76i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-2.22 + 1.18i)T + (37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-1.94 - 2.90i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-7.78 - 5.19i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (7.29 + 3.02i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-10.1 + 0.995i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (0.824 - 4.14i)T + (-82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-6.89 - 6.89i)T + 97iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.72925854164215417794937844204, −10.12430750311813547065474275223, −8.985490475759418794761949287660, −8.276567325472443093982904258994, −7.06750192436793900939388649301, −6.49000524887478857244908701601, −5.29675945119117204654550746036, −4.80838421770373263300866030793, −2.17121829101047276548733324125, −0.52692626932915373907899239153,
2.30587656074065091168215981302, 3.15264745149238247804719541910, 4.59978163467101299515465258799, 5.75342708537608933161558562611, 6.56904471796140078000647908508, 8.421317998318242541115117452814, 9.096165979071506302144247821060, 10.27760708237411762586172485233, 10.47114222099369674360284398318, 11.12591207157708834368893838703
