| L(s) = 1 | + (1.38 + 0.693i)2-s + (−0.395 − 1.68i)3-s + (0.230 + 0.309i)4-s + (0.145 + 0.487i)5-s + (0.622 − 2.60i)6-s + (1.57 + 3.66i)7-s + (−0.432 − 2.45i)8-s + (−2.68 + 1.33i)9-s + (−0.136 + 0.774i)10-s + (−3.72 + 0.882i)11-s + (0.430 − 0.511i)12-s + (−2.60 − 1.71i)13-s + (−0.358 + 6.15i)14-s + (0.764 − 0.438i)15-s + (1.32 − 4.42i)16-s + (−1.41 + 0.513i)17-s + ⋯ |
| L(s) = 1 | + (0.976 + 0.490i)2-s + (−0.228 − 0.973i)3-s + (0.115 + 0.154i)4-s + (0.0652 + 0.218i)5-s + (0.254 − 1.06i)6-s + (0.597 + 1.38i)7-s + (−0.153 − 0.868i)8-s + (−0.895 + 0.444i)9-s + (−0.0431 + 0.244i)10-s + (−1.12 + 0.266i)11-s + (0.124 − 0.147i)12-s + (−0.721 − 0.474i)13-s + (−0.0957 + 1.64i)14-s + (0.197 − 0.113i)15-s + (0.331 − 1.10i)16-s + (−0.342 + 0.124i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31597 - 0.0315277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31597 - 0.0315277i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.395 + 1.68i)T \) |
| good | 2 | \( 1 + (-1.38 - 0.693i)T + (1.19 + 1.60i)T^{2} \) |
| 5 | \( 1 + (-0.145 - 0.487i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (-1.57 - 3.66i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (3.72 - 0.882i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (2.60 + 1.71i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (1.41 - 0.513i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-6.30 - 2.29i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (0.469 - 1.08i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (0.402 + 6.90i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (-0.0460 + 0.00538i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (2.33 + 1.96i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (5.30 - 2.66i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (0.163 + 0.173i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-4.70 - 0.550i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (-6.81 + 11.7i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.35 + 0.322i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (0.187 - 0.251i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (0.319 - 5.48i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (2.03 - 11.5i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.70 - 15.3i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (12.5 + 6.28i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (-0.628 - 0.315i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (-2.00 - 11.3i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.73 + 12.4i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−14.33616536030967424568468652736, −13.29901290478580017334214120239, −12.43797696690580535315154881155, −11.64502597798988942804293010136, −9.978579595114956712994636291990, −8.310779146987945071580701264622, −7.17672821207726807860917762650, −5.73113946728296859430426074278, −5.15516878839356366364642078156, −2.60671830061848423751900771951,
3.19372032015702834254008307962, 4.59788691654708733956250374459, 5.22657713738130083533720189513, 7.42215377828390986814249495330, 8.916187584545239720920216026675, 10.41044867052756016596362980543, 11.09853943982177264286981252838, 12.17937174441706641612365555010, 13.53878359566477713683573981543, 14.08630145879796316305546029966
