| L(s) = 1 | + (−0.485 − 1.32i)2-s + (−0.755 + 0.654i)3-s + (−1.52 + 1.28i)4-s + (3.17 − 0.456i)5-s + (1.23 + 0.686i)6-s + (0.698 − 1.52i)7-s + (2.45 + 1.40i)8-s + (0.142 − 0.989i)9-s + (−2.14 − 3.99i)10-s + (−0.964 + 3.28i)11-s + (0.311 − 1.97i)12-s + (1.14 − 0.520i)13-s + (−2.36 − 0.185i)14-s + (−2.10 + 2.42i)15-s + (0.676 − 3.94i)16-s + (−2.77 − 1.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.343 − 0.939i)2-s + (−0.436 + 0.378i)3-s + (−0.764 + 0.644i)4-s + (1.42 − 0.204i)5-s + (0.504 + 0.280i)6-s + (0.263 − 0.577i)7-s + (0.867 + 0.497i)8-s + (0.0474 − 0.329i)9-s + (−0.679 − 1.26i)10-s + (−0.290 + 0.990i)11-s + (0.0899 − 0.570i)12-s + (0.316 − 0.144i)13-s + (−0.633 − 0.0496i)14-s + (−0.542 + 0.626i)15-s + (0.169 − 0.985i)16-s + (−0.673 − 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22370 - 0.475765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22370 - 0.475765i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.485 + 1.32i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (-3.44 - 3.33i)T \) |
| good | 5 | \( 1 + (-3.17 + 0.456i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.698 + 1.52i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.964 - 3.28i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.14 + 0.520i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.77 + 1.78i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.60 - 7.16i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.16 + 6.48i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.68 + 6.55i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.57 + 0.226i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.21 + 8.41i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.77 - 2.40i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 + (-4.80 - 2.19i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-10.4 + 4.78i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-10.1 - 8.81i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.87 + 6.37i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (11.8 - 3.47i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-5.28 + 3.39i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.798 - 1.74i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (10.7 + 1.54i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-2.00 - 2.31i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.127 - 0.883i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.39050610944391260299419559122, −9.938647424055662459365945122721, −9.438121437940071954742074405643, −8.240821139438977302323521020080, −7.17357264665457912018953973917, −5.81251159008729351220788900975, −4.95694835223268741891205760487, −3.94431849663195825350507791994, −2.42466814269953186620589777529, −1.24870561248597637798987163139,
1.20628223876411496336097347615, 2.76171211402923985647126503316, 4.96927209167198657394268314637, 5.43248453649337294468573050720, 6.53043906424843940674847405998, 6.84685213510217471220575282958, 8.494376144559940931695999497852, 8.811965034977013909821655739950, 9.934881264205367383329976150622, 10.70653746646579371037515782084
