L(s) = 1 | + (1.22 − 0.835i)2-s + (0.435 + 1.67i)3-s + (0.0738 − 0.188i)4-s + (−0.839 − 3.67i)5-s + (1.93 + 1.69i)6-s + (2.50 + 0.847i)7-s + (0.593 + 2.60i)8-s + (−2.62 + 1.45i)9-s + (−4.10 − 3.80i)10-s + (3.06 + 1.47i)11-s + (0.347 + 0.0419i)12-s + (3.79 − 2.58i)13-s + (3.78 − 1.05i)14-s + (5.80 − 3.00i)15-s + (3.19 + 2.96i)16-s + (−1.28 − 3.26i)17-s + ⋯ |
L(s) = 1 | + (0.867 − 0.591i)2-s + (0.251 + 0.967i)3-s + (0.0369 − 0.0941i)4-s + (−0.375 − 1.64i)5-s + (0.790 + 0.690i)6-s + (0.947 + 0.320i)7-s + (0.209 + 0.919i)8-s + (−0.873 + 0.486i)9-s + (−1.29 − 1.20i)10-s + (0.922 + 0.444i)11-s + (0.100 + 0.0120i)12-s + (1.05 − 0.717i)13-s + (1.01 − 0.282i)14-s + (1.49 − 0.776i)15-s + (0.799 + 0.742i)16-s + (−0.310 − 0.791i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.39474 - 0.155835i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.39474 - 0.155835i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.435 - 1.67i)T \) |
| 7 | \( 1 + (-2.50 - 0.847i)T \) |
good | 2 | \( 1 + (-1.22 + 0.835i)T + (0.730 - 1.86i)T^{2} \) |
| 5 | \( 1 + (0.839 + 3.67i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (-3.06 - 1.47i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-3.79 + 2.58i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (1.28 + 3.26i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-2.34 - 4.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.10 + 3.89i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (2.21 + 0.334i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (-2.86 - 4.95i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.60 + 1.29i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (3.21 + 0.991i)T + (33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (2.48 - 0.766i)T + (35.5 - 24.2i)T^{2} \) |
| 47 | \( 1 + (-5.44 + 3.70i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (4.51 - 0.680i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (9.08 - 2.80i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (1.82 + 4.65i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (7.14 + 12.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.88 - 7.37i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.305 - 4.07i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (0.616 - 1.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.61 + 3.14i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (1.11 + 0.761i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (6.47 + 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.40519683492644517187224454850, −10.39236289997245572731629733836, −9.091454818998404820627860934419, −8.548494249412090292339794121039, −7.915185405955900921772588557604, −5.68517528128006089943614507031, −4.93757802699816033681010089010, −4.30974237300475436332442904681, −3.43813557299664027625600121492, −1.68297675061043086794280840049,
1.59825463949783999981701693094, 3.33837011245404330399214003601, 4.13260710883046464146370816895, 5.81094998608975659525091472432, 6.53700954218460569976062277492, 7.11253507186003871788263184561, 7.987234122343065291474162212144, 9.147809110386615579267881875602, 10.57290114885147665402869275086, 11.37594876013224038819175764284