L(s) = 1 | + (−0.309 + 0.951i)3-s + 3.54·7-s + (−0.809 − 0.587i)9-s + (1.78 − 1.29i)11-s + (−5.80 − 4.21i)13-s + (−1.96 − 6.05i)17-s + (−0.715 − 2.20i)19-s + (−1.09 + 3.37i)21-s + (1.76 − 1.27i)23-s + (0.809 − 0.587i)27-s + (0.262 − 0.806i)29-s + (−1.32 − 4.09i)31-s + (0.681 + 2.09i)33-s + (−5.84 − 4.24i)37-s + (5.80 − 4.21i)39-s + ⋯ |
L(s) = 1 | + (−0.178 + 0.549i)3-s + 1.34·7-s + (−0.269 − 0.195i)9-s + (0.538 − 0.390i)11-s + (−1.61 − 1.17i)13-s + (−0.477 − 1.46i)17-s + (−0.164 − 0.504i)19-s + (−0.239 + 0.736i)21-s + (0.367 − 0.266i)23-s + (0.155 − 0.113i)27-s + (0.0486 − 0.149i)29-s + (−0.238 − 0.734i)31-s + (0.118 + 0.365i)33-s + (−0.961 − 0.698i)37-s + (0.929 − 0.675i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.361069258\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.361069258\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.54T + 7T^{2} \) |
| 11 | \( 1 + (-1.78 + 1.29i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (5.80 + 4.21i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.96 + 6.05i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.715 + 2.20i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.76 + 1.27i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.262 + 0.806i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.32 + 4.09i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.84 + 4.24i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 0.790i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.18T + 43T^{2} \) |
| 47 | \( 1 + (1.87 - 5.75i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.69 - 11.3i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-10.0 - 7.33i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.59 + 4.06i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.46 + 4.50i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-4.25 + 13.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.881 + 0.640i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.80 - 5.55i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.87 + 11.9i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.68 + 4.12i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.60 + 17.2i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.335498922488828574596123047321, −8.616142446969114771614995280078, −7.68156718166447862718861350071, −7.09608368661898267295732479447, −5.83047338814991533252837574543, −4.88133278956454786836048945571, −4.66419044306115219272915240507, −3.20392189191097730213599787326, −2.21817686805930054959327252153, −0.53842302205453810493098686846,
1.62169878947702351106626197799, 2.08254129811286889723828640489, 3.78772624508900887291063389522, 4.74866832136389623725729725300, 5.35081670362626921793069892776, 6.73601347823855409725545155761, 7.00209795459437071349623182439, 8.165945575922864537343910518379, 8.566157030517854781458068200312, 9.668391065587528623632143663416