L(s) = 1 | − 0.699·5-s + (−0.461 − 2.60i)7-s − 0.265i·11-s + (−1.13 + 0.657i)13-s + (−1.86 − 3.22i)17-s + (−0.382 − 0.220i)19-s − 4.96i·23-s − 4.51·25-s + (0.273 + 0.157i)29-s + (−4.85 − 2.80i)31-s + (0.322 + 1.82i)35-s + (−0.351 + 0.608i)37-s + (−5.39 − 9.34i)41-s + (3.73 − 6.46i)43-s + (3.50 + 6.06i)47-s + ⋯ |
L(s) = 1 | − 0.312·5-s + (−0.174 − 0.984i)7-s − 0.0799i·11-s + (−0.315 + 0.182i)13-s + (−0.452 − 0.783i)17-s + (−0.0877 − 0.0506i)19-s − 1.03i·23-s − 0.902·25-s + (0.0507 + 0.0292i)29-s + (−0.872 − 0.503i)31-s + (0.0545 + 0.308i)35-s + (−0.0577 + 0.0999i)37-s + (−0.842 − 1.45i)41-s + (0.569 − 0.985i)43-s + (0.510 + 0.884i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.443788 - 0.756086i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.443788 - 0.756086i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.461 + 2.60i)T \) |
good | 5 | \( 1 + 0.699T + 5T^{2} \) |
| 11 | \( 1 + 0.265iT - 11T^{2} \) |
| 13 | \( 1 + (1.13 - 0.657i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.86 + 3.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.382 + 0.220i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.96iT - 23T^{2} \) |
| 29 | \( 1 + (-0.273 - 0.157i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.85 + 2.80i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.351 - 0.608i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.39 + 9.34i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.73 + 6.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.50 - 6.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.51 + 4.91i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.73 + 11.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.89 - 2.82i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.97 + 5.14i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 + (6.66 - 3.84i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.698 + 1.20i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.72 - 6.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.59 + 9.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.18 - 5.30i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.12320953860040027332848287928, −9.255101283294269588563323573430, −8.317821282130908727242259640211, −7.31625732983857660096986122422, −6.80774497337970798446368101082, −5.55597548532853086312247591872, −4.43332705425778612427613941561, −3.65902574591780933933021284065, −2.25259846550042582501883753336, −0.42997152524882187893442896042,
1.83483047714311549330654398187, 3.07940968862495292102468230196, 4.19481792827541059342720077185, 5.38238654849316955757298767943, 6.11503905407851908390555937656, 7.21832408943420340908835512018, 8.113701742793539600921084364488, 8.922166927107013255859915964335, 9.699607855726992924669447490516, 10.61815206648131484068813245495