| L(s) = 1 | + (−0.673 + 0.565i)2-s + (0.5 + 0.181i)3-s + (−0.213 + 1.20i)4-s + (−0.439 − 2.49i)5-s + (−0.439 + 0.160i)6-s + (0.939 − 1.62i)7-s + (−1.41 − 2.45i)8-s + (−2.08 − 1.74i)9-s + (1.70 + 1.43i)10-s + (−1.70 − 2.95i)11-s + (−0.326 + 0.565i)12-s + (−4.97 + 1.80i)13-s + (0.286 + 1.62i)14-s + (0.233 − 1.32i)15-s + (0.0393 + 0.0143i)16-s + (1.26 − 1.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.476 + 0.399i)2-s + (0.288 + 0.105i)3-s + (−0.106 + 0.604i)4-s + (−0.196 − 1.11i)5-s + (−0.179 + 0.0653i)6-s + (0.355 − 0.615i)7-s + (−0.501 − 0.868i)8-s + (−0.693 − 0.582i)9-s + (0.539 + 0.452i)10-s + (−0.514 − 0.890i)11-s + (−0.0942 + 0.163i)12-s + (−1.37 + 0.501i)13-s + (0.0767 + 0.434i)14-s + (0.0604 − 0.342i)15-s + (0.00984 + 0.00358i)16-s + (0.307 − 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0996 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0996 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.503878 - 0.455940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.503878 - 0.455940i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (0.673 - 0.565i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.5 - 0.181i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.439 + 2.49i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 1.62i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.97 - 1.80i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 1.06i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.305 + 1.73i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.65 - 2.22i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.971 + 1.68i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.837T + 37T^{2} \) |
| 41 | \( 1 + (4.21 + 1.53i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.833 - 4.72i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.549 - 0.460i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.06 + 6.01i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-8.24 + 6.91i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.762 + 4.32i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.8 + 9.13i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.38 - 13.5i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-7.06 - 2.57i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (6.54 + 2.38i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.25 - 2.17i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.14 - 0.780i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.39 + 1.17i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.38130724279882100326184860030, −9.967064210340741790274309322669, −9.108005252771476523505548203087, −8.403755273115356855862168655306, −7.76938508539912574445737438117, −6.67578181480919459274024482752, −5.21441850036702268605367884187, −4.18221103806206400766652075431, −2.92336255440384690723824201808, −0.50050940851085495379485287330,
2.18459362878751423014311518926, 2.84230467165388876481001056480, 4.90638691358766911495632771715, 5.72055837715327194058806934380, 7.13421593270805213482529161696, 7.965536940492688601286998672864, 8.966806136456526660150342043859, 10.12629991064376184297259443785, 10.46544793885519865743651485573, 11.52836854484906191591932642207
