L(s) = 1 | + (1.19 + 2.06i)2-s + (0.266 − 1.71i)3-s + (−1.84 + 3.20i)4-s − 2.92·5-s + (3.85 − 1.49i)6-s + (2.35 − 1.20i)7-s − 4.05·8-s + (−2.85 − 0.913i)9-s + (−3.48 − 6.03i)10-s − 1.35·11-s + (4.98 + 4.01i)12-s + (−0.733 − 1.26i)13-s + (5.29 + 3.43i)14-s + (−0.779 + 4.99i)15-s + (−1.13 − 1.96i)16-s + (1.65 + 2.86i)17-s + ⋯ |
L(s) = 1 | + (0.843 + 1.46i)2-s + (0.154 − 0.988i)3-s + (−0.924 + 1.60i)4-s − 1.30·5-s + (1.57 − 0.608i)6-s + (0.891 − 0.453i)7-s − 1.43·8-s + (−0.952 − 0.304i)9-s + (−1.10 − 1.90i)10-s − 0.408·11-s + (1.43 + 1.16i)12-s + (−0.203 − 0.352i)13-s + (1.41 + 0.919i)14-s + (−0.201 + 1.29i)15-s + (−0.284 − 0.492i)16-s + (0.401 + 0.695i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.982468 + 0.600807i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.982468 + 0.600807i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.266 + 1.71i)T \) |
| 7 | \( 1 + (-2.35 + 1.20i)T \) |
good | 2 | \( 1 + (-1.19 - 2.06i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + (0.733 + 1.26i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.65 - 2.86i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.10 - 1.91i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + (-0.521 + 0.903i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.63 - 2.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.43 + 9.41i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.904 + 1.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.17 - 3.76i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.98 + 3.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.22 + 5.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.279 + 0.484i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.40 - 11.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + (-5.22 - 9.05i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.383 + 0.664i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.983 - 1.70i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.20 + 5.54i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.14 - 7.17i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.87948190725450317951336938574, −14.43285985099618476812732854176, −13.14529526093128969126645940228, −12.32732877707333912346962592835, −11.08241246194059462622958042965, −8.279779662982566325496196472502, −7.85854727084343000468233115495, −6.90715392699090883111506020060, −5.35404388869163689800683830451, −3.84184665474636678764671848242,
2.89378383945495108507831119830, 4.28790500631972028391276575958, 5.11397260084066045235725927060, 7.978321057653895087580353550234, 9.370663155108345699689735412055, 10.74637762699573106940765325429, 11.47210459514479716300383983776, 12.11722546718272943896380855097, 13.63233750479999278852533229178, 14.78067739915754840144998284638