L(s) = 1 | + (−17.1 + 17.1i)7-s + 68.0i·11-s + (33.1 + 33.1i)13-s + (−15.5 − 15.5i)17-s + 40.1i·19-s + (52.1 − 52.1i)23-s + 97.9·29-s − 206.·31-s + (238. − 238. i)37-s − 43.1i·41-s + (−255. − 255. i)43-s + (−134. − 134. i)47-s − 248. i·49-s + (−458. + 458. i)53-s − 748.·59-s + ⋯ |
L(s) = 1 | + (−0.928 + 0.928i)7-s + 1.86i·11-s + (0.707 + 0.707i)13-s + (−0.222 − 0.222i)17-s + 0.484i·19-s + (0.473 − 0.473i)23-s + 0.627·29-s − 1.19·31-s + (1.05 − 1.05i)37-s − 0.164i·41-s + (−0.905 − 0.905i)43-s + (−0.418 − 0.418i)47-s − 0.724i·49-s + (−1.18 + 1.18i)53-s − 1.65·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7252689384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7252689384\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (17.1 - 17.1i)T - 343iT^{2} \) |
| 11 | \( 1 - 68.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-33.1 - 33.1i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (15.5 + 15.5i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 - 40.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-52.1 + 52.1i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 97.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-238. + 238. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 43.1iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (255. + 255. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (134. + 134. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (458. - 458. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 748.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 240.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (133. - 133. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 899. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (26.5 + 26.5i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 112. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (644. - 644. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 554.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.24e3 + 1.24e3i)T - 9.12e5iT^{2} \) |
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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
−9.968786781233269479707333936115, −9.339640786661826042343335417998, −8.726564689760703294054116973020, −7.48733491809741411799276656207, −6.73091833047101448598294376883, −5.94585742060679880344260336712, −4.85238395417734512526168988865, −3.90242906659273310501323475729, −2.65444924713947472197257396130, −1.70032246664575322560969828214,
0.20029059867829206105051361313, 1.15685524573008074933237484902, 3.15828481397898227107454719676, 3.45251098295780027560873215492, 4.82315684866160162344080637909, 6.05643595494048267243391826560, 6.47930086958755028800474743740, 7.66762059864145797301416535889, 8.428358172347610281232858781101, 9.284945161617975574299712941166