L(s) = 1 | + 3.61·5-s − i·7-s + 2.00i·11-s + 1.59i·13-s + 7.79i·17-s + 5.05·19-s − 7.14·23-s + 8.03·25-s + 4.94·29-s + 2.53i·31-s − 3.61i·35-s + 4.76i·37-s − 0.836i·41-s − 0.151·43-s − 0.795·47-s + ⋯ |
L(s) = 1 | + 1.61·5-s − 0.377i·7-s + 0.605i·11-s + 0.441i·13-s + 1.88i·17-s + 1.15·19-s − 1.49·23-s + 1.60·25-s + 0.917·29-s + 0.454i·31-s − 0.610i·35-s + 0.783i·37-s − 0.130i·41-s − 0.0230·43-s − 0.116·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.630711116\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.630711116\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 11 | \( 1 - 2.00iT - 11T^{2} \) |
| 13 | \( 1 - 1.59iT - 13T^{2} \) |
| 17 | \( 1 - 7.79iT - 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 37 | \( 1 - 4.76iT - 37T^{2} \) |
| 41 | \( 1 + 0.836iT - 41T^{2} \) |
| 43 | \( 1 + 0.151T + 43T^{2} \) |
| 47 | \( 1 + 0.795T + 47T^{2} \) |
| 53 | \( 1 - 1.05T + 53T^{2} \) |
| 59 | \( 1 + 8.34iT - 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 + 0.655T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 3.81iT - 79T^{2} \) |
| 83 | \( 1 - 14.9iT - 83T^{2} \) |
| 89 | \( 1 + 0.0444iT - 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−8.273538775212419132438981516322, −7.44973862830171642492640722700, −6.59046201772582759370033347175, −6.12002040035411694765267477743, −5.48650398533776501641467653698, −4.62844437224244174726224388692, −3.85131638942253586138738652031, −2.80129918553484670825594073639, −1.83457163858762806387426647899, −1.35900259238714226761576373542,
0.63407797298917511153003753469, 1.74190331478838623891951581651, 2.65231534152723635057183922523, 3.14775117189998346708542551823, 4.48388203199600231545949287199, 5.32191882986628644499690300588, 5.74669530318515127967337105438, 6.32329410034360214530431383649, 7.22842662121648961685697534538, 7.894335755418619863768119985945