| L(s) = 1 | + (−0.404 + 0.467i)2-s + (−1.88 + 1.20i)3-s + (0.230 + 1.60i)4-s + (0.513 + 1.12i)5-s + (0.196 − 1.36i)6-s + (−3.10 + 0.911i)7-s + (−1.88 − 1.20i)8-s + (0.830 − 1.81i)9-s + (−0.732 − 0.215i)10-s + (3.42 + 3.95i)11-s + (−2.36 − 2.73i)12-s + (−2.87 − 0.845i)13-s + (0.830 − 1.81i)14-s + (−2.32 − 1.49i)15-s + (−1.77 + 0.522i)16-s + (−0.108 + 0.756i)17-s + ⋯ |
| L(s) = 1 | + (−0.286 + 0.330i)2-s + (−1.08 + 0.697i)3-s + (0.115 + 0.800i)4-s + (0.229 + 0.502i)5-s + (0.0802 − 0.558i)6-s + (−1.17 + 0.344i)7-s + (−0.665 − 0.427i)8-s + (0.276 − 0.606i)9-s + (−0.231 − 0.0680i)10-s + (1.03 + 1.19i)11-s + (−0.683 − 0.789i)12-s + (−0.798 − 0.234i)13-s + (0.222 − 0.486i)14-s + (−0.600 − 0.385i)15-s + (−0.444 + 0.130i)16-s + (−0.0263 + 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.219128 - 0.360524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.219128 - 0.360524i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (0.404 - 0.467i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (1.88 - 1.20i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.513 - 1.12i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (3.10 - 0.911i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-3.42 - 3.95i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.87 + 0.845i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.108 - 0.756i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.284 - 1.97i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.426 + 2.96i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.64 - 3.62i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.513 - 1.12i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.44 + 3.15i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + (0.453 - 0.133i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (6.20 + 1.82i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (5.84 + 3.75i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.80 + 2.08i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (8.01 - 9.24i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.929 + 6.46i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-10.5 - 3.08i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (3.64 - 7.97i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (8.80 - 5.66i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.35 - 16.1i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.54038236549392098401501011176, −10.28585713856470354213631410165, −9.842217227130298076872894302400, −9.015151439052344413529936738655, −7.69352074834862165595983980745, −6.57433688362720852410047592563, −6.33804353877344323224118908608, −4.89725193216967708194115208260, −3.84876495836599575392485304277, −2.62901373073640990391455190089,
0.32970530904491002570103110089, 1.32409595106838190143340645655, 3.08911491454432118587413003199, 4.79902056937899413589265602606, 5.88324129654862608979327686360, 6.38775485774409097569900660945, 7.15162749916483381993030383198, 8.842001496726690090075718457864, 9.438496360116266740300727883430, 10.31447780057071984535835774041
