| L(s) = 1 | + (0.461 − 0.681i)2-s + (−0.107 − 1.98i)3-s + (0.489 + 1.22i)4-s + (−2.53 − 1.17i)5-s + (−1.40 − 0.845i)6-s + (0.992 + 3.57i)7-s + (2.67 + 0.587i)8-s + (−0.960 + 0.104i)9-s + (−1.96 + 1.18i)10-s + (−1.59 − 1.51i)11-s + (2.39 − 1.10i)12-s + (0.353 + 0.0384i)13-s + (2.89 + 0.975i)14-s + (−2.05 + 5.16i)15-s + (−0.285 + 0.270i)16-s + (−1.46 + 5.27i)17-s + ⋯ |
| L(s) = 1 | + (0.326 − 0.481i)2-s + (−0.0622 − 1.14i)3-s + (0.244 + 0.614i)4-s + (−1.13 − 0.524i)5-s + (−0.573 − 0.345i)6-s + (0.375 + 1.35i)7-s + (0.944 + 0.207i)8-s + (−0.320 + 0.0348i)9-s + (−0.622 + 0.374i)10-s + (−0.481 − 0.456i)11-s + (0.690 − 0.319i)12-s + (0.0980 + 0.0106i)13-s + (0.773 + 0.260i)14-s + (−0.531 + 1.33i)15-s + (−0.0714 + 0.0676i)16-s + (−0.355 + 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.833078 - 0.455935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.833078 - 0.455935i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (0.644 + 7.65i)T \) |
| good | 2 | \( 1 + (-0.461 + 0.681i)T + (-0.740 - 1.85i)T^{2} \) |
| 3 | \( 1 + (0.107 + 1.98i)T + (-2.98 + 0.324i)T^{2} \) |
| 5 | \( 1 + (2.53 + 1.17i)T + (3.23 + 3.81i)T^{2} \) |
| 7 | \( 1 + (-0.992 - 3.57i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (1.59 + 1.51i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.353 - 0.0384i)T + (12.6 + 2.79i)T^{2} \) |
| 17 | \( 1 + (1.46 - 5.27i)T + (-14.5 - 8.76i)T^{2} \) |
| 19 | \( 1 + (5.52 + 4.19i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (1.69 + 3.18i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (-1.55 - 2.30i)T + (-10.7 + 26.9i)T^{2} \) |
| 31 | \( 1 + (-6.43 + 4.89i)T + (8.29 - 29.8i)T^{2} \) |
| 37 | \( 1 + (-5.95 + 1.30i)T + (33.5 - 15.5i)T^{2} \) |
| 41 | \( 1 + (-1.50 + 2.83i)T + (-23.0 - 33.9i)T^{2} \) |
| 43 | \( 1 + (8.24 - 7.81i)T + (2.32 - 42.9i)T^{2} \) |
| 47 | \( 1 + (2.11 - 0.980i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (-4.72 - 2.84i)T + (24.8 + 46.8i)T^{2} \) |
| 61 | \( 1 + (2.34 - 3.46i)T + (-22.5 - 56.6i)T^{2} \) |
| 67 | \( 1 + (-6.24 - 1.37i)T + (60.8 + 28.1i)T^{2} \) |
| 71 | \( 1 + (-4.69 + 2.17i)T + (45.9 - 54.1i)T^{2} \) |
| 73 | \( 1 + (-2.76 - 0.932i)T + (58.1 + 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.208 + 3.85i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (1.60 - 9.80i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (0.215 + 0.317i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (-14.9 + 5.03i)T + (77.2 - 58.7i)T^{2} \) |
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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
−15.06641210217308360971659537660, −13.23801860500440244894616750816, −12.61308897715958800206767660018, −11.91785736381960103076985603413, −11.03292824958249228767472938524, −8.394361934127243216576946287853, −8.093590138544953339185138973985, −6.42823315721419072822343232686, −4.40143539135314726633799486339, −2.36607353881058965220449846812,
3.97488039960312470399056557832, 4.83205289313142761377890700059, 6.84770454090659688403904271148, 7.84585479755973246028210126956, 10.00605634881785132726976116305, 10.59123213715448655655924597269, 11.55538016209177320507648637419, 13.53636671197974836858235486937, 14.59616695134797296325722765046, 15.38716349555220217798244430283
