| L(s) = 1 | + (1.02 + 0.451i)2-s + (2.39 + 1.40i)3-s + (−0.507 − 0.556i)4-s + (−1.83 + 2.10i)5-s + (1.81 + 2.51i)6-s + (1.50 + 1.18i)7-s + (−0.976 − 2.91i)8-s + (2.29 + 4.12i)9-s + (−2.81 + 1.32i)10-s + (1.21 − 0.439i)11-s + (−0.431 − 2.04i)12-s + (−2.08 − 1.03i)13-s + (0.997 + 1.89i)14-s + (−7.34 + 2.46i)15-s + (0.177 − 1.91i)16-s + (2.17 − 3.49i)17-s + ⋯ |
| L(s) = 1 | + (0.722 + 0.318i)2-s + (1.38 + 0.812i)3-s + (−0.253 − 0.278i)4-s + (−0.818 + 0.941i)5-s + (0.739 + 1.02i)6-s + (0.567 + 0.449i)7-s + (−0.345 − 1.03i)8-s + (0.765 + 1.37i)9-s + (−0.891 + 0.418i)10-s + (0.367 − 0.132i)11-s + (−0.124 − 0.590i)12-s + (−0.578 − 0.288i)13-s + (0.266 + 0.505i)14-s + (−1.89 + 0.635i)15-s + (0.0444 − 0.479i)16-s + (0.528 − 0.848i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89830 + 1.30819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89830 + 1.30819i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-2.17 + 3.49i)T \) |
| good | 2 | \( 1 + (-1.02 - 0.451i)T + (1.34 + 1.47i)T^{2} \) |
| 3 | \( 1 + (-2.39 - 1.40i)T + (1.45 + 2.62i)T^{2} \) |
| 5 | \( 1 + (1.83 - 2.10i)T + (-0.690 - 4.95i)T^{2} \) |
| 7 | \( 1 + (-1.50 - 1.18i)T + (1.60 + 6.81i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 0.439i)T + (8.46 - 7.02i)T^{2} \) |
| 13 | \( 1 + (2.08 + 1.03i)T + (7.83 + 10.3i)T^{2} \) |
| 19 | \( 1 + (-1.06 - 2.42i)T + (-12.8 + 14.0i)T^{2} \) |
| 23 | \( 1 + (-1.21 + 1.53i)T + (-5.26 - 22.3i)T^{2} \) |
| 29 | \( 1 + (6.27 + 2.94i)T + (18.5 + 22.3i)T^{2} \) |
| 31 | \( 1 + (-0.759 + 10.9i)T + (-30.7 - 4.28i)T^{2} \) |
| 37 | \( 1 + (-1.37 + 0.897i)T + (14.9 - 33.8i)T^{2} \) |
| 41 | \( 1 + (6.89 - 1.79i)T + (35.8 - 19.9i)T^{2} \) |
| 43 | \( 1 + (-9.14 - 7.59i)T + (7.90 + 42.2i)T^{2} \) |
| 47 | \( 1 + (-5.91 + 1.68i)T + (39.9 - 24.7i)T^{2} \) |
| 53 | \( 1 + (9.44 - 5.26i)T + (27.9 - 45.0i)T^{2} \) |
| 59 | \( 1 + (0.780 - 3.31i)T + (-52.8 - 26.2i)T^{2} \) |
| 61 | \( 1 + (5.16 - 3.71i)T + (19.3 - 57.8i)T^{2} \) |
| 67 | \( 1 + (-3.17 - 1.22i)T + (49.5 + 45.1i)T^{2} \) |
| 71 | \( 1 + (-0.0387 - 0.333i)T + (-69.1 + 16.2i)T^{2} \) |
| 73 | \( 1 + (-0.692 + 0.214i)T + (60.2 - 41.2i)T^{2} \) |
| 79 | \( 1 + (-2.75 - 2.62i)T + (3.64 + 78.9i)T^{2} \) |
| 83 | \( 1 + (7.63 - 1.06i)T + (79.8 - 22.7i)T^{2} \) |
| 89 | \( 1 + (7.44 - 3.70i)T + (53.6 - 71.0i)T^{2} \) |
| 97 | \( 1 + (-11.1 + 1.29i)T + (94.4 - 22.2i)T^{2} \) |
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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
−12.01204608269674321921812773679, −11.02189214356996106781235341208, −9.830682192888916509909066957966, −9.303582474962487487022962931417, −8.036848674889733921187982713856, −7.33685165973323976325031495708, −5.79239744310906348823569940181, −4.53139412494152784143660092973, −3.70411166842675985265284084708, −2.70993854762283968929417882480,
1.63530035067986513585336616054, 3.22127552004908582586826098150, 4.11633535342705962407284927507, 5.11422004772713632718547083913, 7.13611000247470254934969743724, 7.896883712295155517506788625864, 8.603322013750200539189459134313, 9.271659778345512067068982999609, 11.02328437321615276238806115288, 12.31983314501180963569232067306
