| L(s) = 1 | + (0.978 + 0.207i)2-s + (−0.450 − 1.38i)3-s + (0.913 + 0.406i)4-s + (−0.148 − 1.41i)5-s + (−0.152 − 1.44i)6-s + (0.837 + 0.929i)7-s + (0.809 + 0.587i)8-s + (0.709 − 0.515i)9-s + (0.148 − 1.41i)10-s − 1.58·11-s + (0.152 − 1.44i)12-s + (−2.12 + 3.68i)13-s + (0.625 + 1.08i)14-s + (−1.89 + 0.843i)15-s + (0.669 + 0.743i)16-s + (0.780 + 0.347i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (−0.259 − 0.799i)3-s + (0.456 + 0.203i)4-s + (−0.0665 − 0.633i)5-s + (−0.0621 − 0.591i)6-s + (0.316 + 0.351i)7-s + (0.286 + 0.207i)8-s + (0.236 − 0.171i)9-s + (0.0470 − 0.447i)10-s − 0.478·11-s + (0.0439 − 0.418i)12-s + (−0.590 + 1.02i)13-s + (0.167 + 0.289i)14-s + (−0.489 + 0.217i)15-s + (0.167 + 0.185i)16-s + (0.189 + 0.0842i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36541 - 0.387310i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36541 - 0.387310i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 61 | \( 1 + (6.11 - 4.85i)T \) |
| good | 3 | \( 1 + (0.450 + 1.38i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (0.148 + 1.41i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (-0.837 - 0.929i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 + (2.12 - 3.68i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.780 - 0.347i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (4.30 - 4.78i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.20 - 0.876i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 1.92i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.50 - 0.320i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-0.724 + 2.22i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.27 + 10.0i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (4.12 - 1.83i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (1.77 + 3.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.03 - 5.83i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.31 - 0.704i)T + (53.8 + 23.9i)T^{2} \) |
| 67 | \( 1 + (0.193 + 1.83i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-1.38 + 13.1i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (0.792 - 7.54i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-11.5 + 5.15i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (7.62 + 1.61i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (4.56 + 14.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.98 + 0.420i)T + (88.6 - 39.4i)T^{2} \) |
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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
−13.16052939525339087453622913931, −12.35169705401972700484239452195, −11.87759279764944532539452402860, −10.42211905435335691224831330537, −8.904725952825437514023847081792, −7.71087765704379388562120899381, −6.64013627169066578114675677962, −5.44651981588811402221642093564, −4.16308869528855605486254525896, −1.94593226540908128273275458791,
2.79694901877582371163964254054, 4.34427220917274245636892834740, 5.28744494373836993492296133174, 6.80418313404740011565306379447, 7.977503961201701697223338291209, 9.794372506189088626692876202555, 10.63767924619418210506612171115, 11.20653977242235960294592153272, 12.64478218056459693748163621135, 13.47881657777117475268122973171
