| L(s) = 1 | + (2.28 + 0.171i)2-s + (0.623 + 1.61i)3-s + (3.23 + 0.487i)4-s + (0.442 − 1.94i)5-s + (1.14 + 3.80i)6-s + (2.56 + 0.639i)7-s + (2.84 + 0.649i)8-s + (−2.22 + 2.01i)9-s + (1.34 − 4.36i)10-s + (−1.84 − 3.83i)11-s + (1.22 + 5.53i)12-s + (−4.17 − 0.313i)13-s + (5.76 + 1.90i)14-s + (3.41 − 0.493i)15-s + (0.149 + 0.0460i)16-s + (1.78 − 0.268i)17-s + ⋯ |
| L(s) = 1 | + (1.61 + 0.121i)2-s + (0.359 + 0.933i)3-s + (1.61 + 0.243i)4-s + (0.198 − 0.867i)5-s + (0.469 + 1.55i)6-s + (0.970 + 0.241i)7-s + (1.00 + 0.229i)8-s + (−0.741 + 0.671i)9-s + (0.425 − 1.38i)10-s + (−0.556 − 1.15i)11-s + (0.354 + 1.59i)12-s + (−1.15 − 0.0868i)13-s + (1.54 + 0.508i)14-s + (0.880 − 0.127i)15-s + (0.0373 + 0.0115i)16-s + (0.432 − 0.0651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.51753 + 1.05485i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.51753 + 1.05485i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.623 - 1.61i)T \) |
| 7 | \( 1 + (-2.56 - 0.639i)T \) |
| good | 2 | \( 1 + (-2.28 - 0.171i)T + (1.97 + 0.298i)T^{2} \) |
| 5 | \( 1 + (-0.442 + 1.94i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (1.84 + 3.83i)T + (-6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (4.17 + 0.313i)T + (12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (-1.78 + 0.268i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (5.84 - 3.37i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.85 - 3.07i)T + (5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-0.822 - 0.322i)T + (21.2 + 19.7i)T^{2} \) |
| 31 | \( 1 + (8.22 - 4.75i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.25 + 5.74i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-7.66 - 7.11i)T + (3.06 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-5.11 + 4.74i)T + (3.21 - 42.8i)T^{2} \) |
| 47 | \( 1 + (-0.524 + 6.99i)T + (-46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (-0.0150 + 0.00589i)T + (38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (5.74 - 5.33i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (1.68 + 11.1i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (0.481 + 0.834i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.93 - 7.12i)T + (15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-6.90 - 10.1i)T + (-26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-4.55 + 7.89i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.292 + 3.90i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (0.526 + 7.02i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-0.607 + 0.350i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.24085227996499816074523015482, −10.70435485534259408229413115072, −9.299646686899793048987194439331, −8.506494232114476450057694399260, −7.50742312142927045670456630245, −5.76765943457437726992923378951, −5.26391374235296838868115873887, −4.57774170298412163521477437784, −3.50311392152984415955647445241, −2.30010474074204168526520031512,
2.18755010495797983878220604729, 2.67836101963959008945646088676, 4.27264377277371880619905454603, 5.12350798009212843016360207213, 6.34194056097290939058937157965, 7.14013432390782101807817193632, 7.75239065578830324919641066547, 9.234118488660144574791951959084, 10.68397110694669700903157184166, 11.21121017165374358621869415265
