| L(s) = 1 | + (−0.342 + 2.57i)3-s + (−0.628 + 0.277i)5-s + (−1.47 − 3.17i)7-s + (−3.60 − 0.977i)9-s + (−0.684 − 0.877i)11-s + (−1.75 − 3.42i)13-s + (−0.499 − 1.71i)15-s + (0.451 − 0.0342i)17-s + (−4.29 − 6.46i)19-s + (8.66 − 2.70i)21-s + (1.87 − 0.745i)23-s + (−3.04 + 3.34i)25-s + (0.736 − 1.75i)27-s + (−0.785 − 0.312i)29-s + (−1.72 − 0.399i)31-s + ⋯ |
| L(s) = 1 | + (−0.197 + 1.48i)3-s + (−0.281 + 0.124i)5-s + (−0.557 − 1.19i)7-s + (−1.20 − 0.325i)9-s + (−0.206 − 0.264i)11-s + (−0.486 − 0.951i)13-s + (−0.128 − 0.442i)15-s + (0.109 − 0.00829i)17-s + (−0.985 − 1.48i)19-s + (1.88 − 0.590i)21-s + (0.391 − 0.155i)23-s + (−0.609 + 0.669i)25-s + (0.141 − 0.337i)27-s + (−0.145 − 0.0579i)29-s + (−0.310 − 0.0717i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.382252 - 0.327604i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.382252 - 0.327604i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (-9.42 + 8.84i)T \) |
| good | 3 | \( 1 + (0.342 - 2.57i)T + (-2.89 - 0.785i)T^{2} \) |
| 5 | \( 1 + (0.628 - 0.277i)T + (3.36 - 3.69i)T^{2} \) |
| 7 | \( 1 + (1.47 + 3.17i)T + (-4.51 + 5.35i)T^{2} \) |
| 11 | \( 1 + (0.684 + 0.877i)T + (-2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (1.75 + 3.42i)T + (-7.60 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.451 + 0.0342i)T + (16.8 - 2.56i)T^{2} \) |
| 19 | \( 1 + (4.29 + 6.46i)T + (-7.35 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.87 + 0.745i)T + (16.7 - 15.7i)T^{2} \) |
| 29 | \( 1 + (0.785 + 0.312i)T + (21.0 + 19.9i)T^{2} \) |
| 31 | \( 1 + (1.72 + 0.399i)T + (27.8 + 13.6i)T^{2} \) |
| 37 | \( 1 + (3.03 - 0.823i)T + (31.9 - 18.7i)T^{2} \) |
| 41 | \( 1 + (-8.80 - 4.29i)T + (25.2 + 32.3i)T^{2} \) |
| 43 | \( 1 + (-0.281 - 2.96i)T + (-42.2 + 8.08i)T^{2} \) |
| 47 | \( 1 + (-0.207 + 10.9i)T + (-46.9 - 1.77i)T^{2} \) |
| 53 | \( 1 + (-1.65 + 9.63i)T + (-49.9 - 17.7i)T^{2} \) |
| 59 | \( 1 + (11.8 + 0.896i)T + (58.3 + 8.89i)T^{2} \) |
| 61 | \( 1 + (-10.0 - 10.2i)T + (-1.15 + 60.9i)T^{2} \) |
| 67 | \( 1 + (4.03 + 1.78i)T + (45.0 + 49.5i)T^{2} \) |
| 71 | \( 1 + (8.69 + 0.991i)T + (69.1 + 15.9i)T^{2} \) |
| 73 | \( 1 + (1.19 + 0.892i)T + (20.4 + 70.0i)T^{2} \) |
| 79 | \( 1 + (2.45 + 6.52i)T + (-59.4 + 52.0i)T^{2} \) |
| 83 | \( 1 + (1.50 + 2.08i)T + (-26.2 + 78.7i)T^{2} \) |
| 89 | \( 1 + (-1.02 + 3.52i)T + (-75.0 - 47.8i)T^{2} \) |
| 97 | \( 1 + (2.71 - 0.626i)T + (87.1 - 42.5i)T^{2} \) |
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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
−10.43015388734673305011175055264, −9.662703231992161313259058851882, −8.842365369270836071912073919252, −7.67020123245358281823304683681, −6.80252975825968121215898995417, −5.54553220004420202184129451878, −4.62851637507661783513273692604, −3.82712979452101853705565096509, −2.93582065916221670748103319820, −0.26343445350389957571545875738,
1.73476073887647222870780336434, 2.60812120002550306865056263708, 4.20476573167986679405300081107, 5.69840942568320793464410375452, 6.23783280290668653436406780190, 7.19686264152497770178414711092, 7.960310978547250267133865750941, 8.839624835183244346929901238078, 9.688002083817141455334094933098, 10.91572311642657431522000959958
