L(s) = 1 | − 2.73·3-s + 4.73i·7-s + 4.46·9-s − 3.46i·11-s + 3.46·13-s − 3.46i·17-s − 2i·19-s − 12.9i·21-s + 2.19i·23-s − 3.99·27-s − 2.53·31-s + 9.46i·33-s − 6·37-s − 9.46·39-s + 9.46·41-s + ⋯ |
L(s) = 1 | − 1.57·3-s + 1.78i·7-s + 1.48·9-s − 1.04i·11-s + 0.960·13-s − 0.840i·17-s − 0.458i·19-s − 2.82i·21-s + 0.457i·23-s − 0.769·27-s − 0.455·31-s + 1.64i·33-s − 0.986·37-s − 1.51·39-s + 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9064526098\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9064526098\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 4.73iT - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 2.19iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 0.196T + 43T^{2} \) |
| 47 | \( 1 - 2.19iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 0.928iT - 61T^{2} \) |
| 67 | \( 1 + 0.196T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 6.39iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3iT - 97T^{2} \) |
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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
−9.341346696471835244945763372174, −8.930876048124846572319040778156, −7.952299432680398701348327371362, −6.74472405175734288215549941783, −6.05629870827409417482628573842, −5.52903270288770044722426606934, −4.97768922843258220212581866855, −3.58631853016481276828094017169, −2.39130949933524008626340830293, −0.870757098151957887564925037967,
0.63697300345061816541401076135, 1.66948858401971837384382269558, 3.76903046843412352088818617593, 4.26845184809000422698765713610, 5.19569184620023023857101114300, 6.13545737207515525797784285937, 6.81879842586343325721072894057, 7.39728205436452180731208985992, 8.357782858638802070712697278761, 9.680857412858175445327760307917