L(s) = 1 | − 3-s − 1.33i·7-s + 9-s − 2.94i·11-s − 2.04·13-s + 3.61i·17-s + 5.35i·19-s + 1.33i·21-s + 8.59i·23-s − 27-s − 5.26i·29-s + 2.08·31-s + 2.94i·33-s − 6.55·37-s + 2.04·39-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.504i·7-s + 0.333·9-s − 0.887i·11-s − 0.566·13-s + 0.876i·17-s + 1.22i·19-s + 0.291i·21-s + 1.79i·23-s − 0.192·27-s − 0.977i·29-s + 0.373·31-s + 0.512i·33-s − 1.07·37-s + 0.326·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.257423235\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.257423235\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.33iT - 7T^{2} \) |
| 11 | \( 1 + 2.94iT - 11T^{2} \) |
| 13 | \( 1 + 2.04T + 13T^{2} \) |
| 17 | \( 1 - 3.61iT - 17T^{2} \) |
| 19 | \( 1 - 5.35iT - 19T^{2} \) |
| 23 | \( 1 - 8.59iT - 23T^{2} \) |
| 29 | \( 1 + 5.26iT - 29T^{2} \) |
| 31 | \( 1 - 2.08T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 + 9.97iT - 47T^{2} \) |
| 53 | \( 1 + 6.12T + 53T^{2} \) |
| 59 | \( 1 - 4.75iT - 59T^{2} \) |
| 61 | \( 1 + 8.51iT - 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 2.62T + 71T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−9.078546233040213713801571507980, −8.043645131557329235431292735998, −7.58777881751358653527267217572, −6.60066555626826331180497649387, −5.82530688320974186596082834773, −5.27913438503734149175477815196, −4.04549777125763973124246757794, −3.53415583092980473157665649614, −2.07685977307812227186517867842, −0.884158086113691994082701455096,
0.61616521460414197998945198697, 2.17345160636381986864073793722, 2.92615821375751669960269595894, 4.47647350117957864787646549868, 4.81229805524614347119843088858, 5.74073714086087992853263206125, 6.73319422291286091196335157319, 7.15301513214336900643467674472, 8.100835345590996190316141029390, 9.177224032386892197289962750086