L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 2.64·7-s − 2.82i·8-s − 12.1i·11-s + 18.5·13-s + 3.74i·14-s + 4.00·16-s + 10.9i·17-s + 20·19-s + 17.1·22-s + 12.1i·23-s + 26.2i·26-s − 5.29·28-s + 41.8i·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.377·7-s − 0.353i·8-s − 1.10i·11-s + 1.42·13-s + 0.267i·14-s + 0.250·16-s + 0.641i·17-s + 1.05·19-s + 0.780·22-s + 0.527i·23-s + 1.01i·26-s − 0.188·28-s + 1.44i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.349480375\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.349480375\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
good | 11 | \( 1 + 12.1iT - 121T^{2} \) |
| 13 | \( 1 - 18.5T + 169T^{2} \) |
| 17 | \( 1 - 10.9iT - 289T^{2} \) |
| 19 | \( 1 - 20T + 361T^{2} \) |
| 23 | \( 1 - 12.1iT - 529T^{2} \) |
| 29 | \( 1 - 41.8iT - 841T^{2} \) |
| 31 | \( 1 - 25.1T + 961T^{2} \) |
| 37 | \( 1 + 38T + 1.36e3T^{2} \) |
| 41 | \( 1 + 60.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 83.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 94.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 15.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 132.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 12.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 76.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 33.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 60.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 4.77iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 188.T + 9.40e3T^{2} \) |
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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
−8.385158565939638143639426385416, −8.076921659038025499843410148259, −6.92721547321301225519257516674, −6.41893469998581887711504928801, −5.46187337568641193834023401869, −5.09699977415403851465787465573, −3.60111384087472555321610825702, −3.47261597652428672824446519397, −1.71606842767150323271330678386, −0.74749394077605292098377474575,
0.827515046007931439759876801546, 1.72264588778647614118880556120, 2.72347773582854611257940365569, 3.64353116302627412737494422427, 4.50788824625749973551207893970, 5.14111018347353378314079462204, 6.14182866941522810651412318008, 6.95337281024950552635033337900, 7.890106759178124587760877278062, 8.445209034311106333897654833136