L(s) = 1 | − 1.41i·5-s − 2·7-s − 4.24i·11-s + 2.82i·13-s + 7.07i·17-s + (1 + 4.24i)19-s − 1.41i·23-s + 2.99·25-s − 10·29-s − 2.82i·31-s + 2.82i·35-s + 5.65i·37-s + 10·41-s − 12·43-s + 1.41i·47-s + ⋯ |
L(s) = 1 | − 0.632i·5-s − 0.755·7-s − 1.27i·11-s + 0.784i·13-s + 1.71i·17-s + (0.229 + 0.973i)19-s − 0.294i·23-s + 0.599·25-s − 1.85·29-s − 0.508i·31-s + 0.478i·35-s + 0.929i·37-s + 1.56·41-s − 1.82·43-s + 0.206i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.280665781\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.280665781\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1 - 4.24i)T \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 2.82iT - 31T^{2} \) |
| 37 | \( 1 - 5.65iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 - 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 11.3iT - 79T^{2} \) |
| 83 | \( 1 + 4.24iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 8.48iT - 97T^{2} \) |
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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
−8.704162875101944710790112934832, −8.453744924674234598200545653154, −7.47618608457496133074704355911, −6.40831483395080228282414480273, −5.97859252613812734879612858794, −5.14355431780296368447505403770, −3.89347823598642255882414816154, −3.56230926850487348522793878462, −2.16636562265307172176770250809, −1.02033610835468270570295208608,
0.48466762490716200556038009173, 2.19305119089552562109018607698, 2.95036380937654436880837791093, 3.78551198529710502008390937481, 4.96257444134477495286624829039, 5.49906396392519123990776075042, 6.79080499329829796541042616820, 7.04931526000035031761977189596, 7.71865709717719150320283949744, 8.906418544850366507911922198744