| L(s) = 1 | + (0.654 − 0.755i)3-s + (−0.0513 + 0.357i)5-s + (1.38 + 3.03i)7-s + (−0.142 − 0.989i)9-s + (5.30 − 1.55i)11-s + (−1.03 + 2.27i)13-s + (0.236 + 0.272i)15-s + (5.44 − 3.49i)17-s + (−6.00 − 3.85i)19-s + (3.19 + 0.938i)21-s + (−4.78 + 0.373i)23-s + (4.67 + 1.37i)25-s + (−0.841 − 0.540i)27-s + (−1.82 + 1.17i)29-s + (−3.06 − 3.54i)31-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.436i)3-s + (−0.0229 + 0.159i)5-s + (0.523 + 1.14i)7-s + (−0.0474 − 0.329i)9-s + (1.60 − 0.469i)11-s + (−0.288 + 0.631i)13-s + (0.0610 + 0.0704i)15-s + (1.32 − 0.848i)17-s + (−1.37 − 0.885i)19-s + (0.697 + 0.204i)21-s + (−0.996 + 0.0778i)23-s + (0.934 + 0.274i)25-s + (−0.161 − 0.104i)27-s + (−0.339 + 0.218i)29-s + (−0.551 − 0.636i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55021 - 0.0324010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55021 - 0.0324010i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (4.78 - 0.373i)T \) |
| good | 5 | \( 1 + (0.0513 - 0.357i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.38 - 3.03i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-5.30 + 1.55i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.03 - 2.27i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.44 + 3.49i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (6.00 + 3.85i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.82 - 1.17i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.06 + 3.54i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.985 - 6.85i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.381 - 2.65i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (8.40 - 9.70i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 2.55T + 47T^{2} \) |
| 53 | \( 1 + (3.84 + 8.42i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.29 + 2.83i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (3.85 + 4.45i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (8.62 + 2.53i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (12.5 + 3.68i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-6.35 - 4.08i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (0.696 - 1.52i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.66 + 11.6i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.59 - 8.76i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.06 + 7.42i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.77204552133598190965527027386, −11.35419677100692923670241624247, −9.696601288625808363397362497930, −8.957619603192607212726782269511, −8.165224000579487958078483424131, −6.88748676857822480803084071403, −6.03633004545929902889431043942, −4.65112971184014238842965674752, −3.14651108741410304709659075049, −1.73705049360753308601186405449,
1.58929971493023225762500145152, 3.71977058559396635918332768059, 4.31638235813868738673674445650, 5.82737005272786546986202292290, 7.13211573744209705810137385009, 8.070601325368895069217203282284, 8.998167584573809330217319185138, 10.26663347658787465239582206427, 10.55959347852500707832431228047, 12.01461161798517594788669883343
