| L(s) = 1 | + (0.735 + 0.793i)2-s + (0.211 − 2.82i)4-s + (−2.21 − 0.867i)5-s + (−1.00 + 6.92i)7-s + (5.77 − 4.60i)8-s + (−0.938 − 2.39i)10-s + (2.23 − 7.23i)11-s + (−3.79 − 16.6i)13-s + (−6.23 + 4.30i)14-s + (−3.28 − 0.495i)16-s + (12.7 + 18.6i)17-s + (−13.1 − 22.6i)19-s + (−2.91 + 6.05i)20-s + (7.37 − 3.55i)22-s + (10.3 − 15.2i)23-s + ⋯ |
| L(s) = 1 | + (0.367 + 0.396i)2-s + (0.0528 − 0.705i)4-s + (−0.442 − 0.173i)5-s + (−0.143 + 0.989i)7-s + (0.722 − 0.575i)8-s + (−0.0938 − 0.239i)10-s + (0.202 − 0.657i)11-s + (−0.291 − 1.27i)13-s + (−0.445 + 0.307i)14-s + (−0.205 − 0.0309i)16-s + (0.749 + 1.09i)17-s + (−0.689 − 1.19i)19-s + (−0.145 + 0.302i)20-s + (0.335 − 0.161i)22-s + (0.452 − 0.662i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.23483 - 1.02317i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23483 - 1.02317i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.00 - 6.92i)T \) |
| good | 2 | \( 1 + (-0.735 - 0.793i)T + (-0.298 + 3.98i)T^{2} \) |
| 5 | \( 1 + (2.21 + 0.867i)T + (18.3 + 17.0i)T^{2} \) |
| 11 | \( 1 + (-2.23 + 7.23i)T + (-99.9 - 68.1i)T^{2} \) |
| 13 | \( 1 + (3.79 + 16.6i)T + (-152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (-12.7 - 18.6i)T + (-105. + 269. i)T^{2} \) |
| 19 | \( 1 + (13.1 + 22.6i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-10.3 + 15.2i)T + (-193. - 492. i)T^{2} \) |
| 29 | \( 1 + (-8.98 + 18.6i)T + (-524. - 657. i)T^{2} \) |
| 31 | \( 1 + (1.98 - 3.43i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (4.62 + 61.7i)T + (-1.35e3 + 204. i)T^{2} \) |
| 41 | \( 1 + (6.35 - 5.06i)T + (374. - 1.63e3i)T^{2} \) |
| 43 | \( 1 + (19.2 - 24.1i)T + (-411. - 1.80e3i)T^{2} \) |
| 47 | \( 1 + (26.3 + 28.4i)T + (-165. + 2.20e3i)T^{2} \) |
| 53 | \( 1 + (-65.1 - 4.87i)T + (2.77e3 + 418. i)T^{2} \) |
| 59 | \( 1 + (-58.1 + 22.8i)T + (2.55e3 - 2.36e3i)T^{2} \) |
| 61 | \( 1 + (-4.11 - 54.9i)T + (-3.67e3 + 554. i)T^{2} \) |
| 67 | \( 1 + (-28.2 + 48.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-56.8 - 117. i)T + (-3.14e3 + 3.94e3i)T^{2} \) |
| 73 | \( 1 + (-49.7 - 46.1i)T + (398. + 5.31e3i)T^{2} \) |
| 79 | \( 1 + (-14.2 - 24.6i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (102. + 23.2i)T + (6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (9.78 + 31.7i)T + (-6.54e3 + 4.46e3i)T^{2} \) |
| 97 | \( 1 - 102.T + 9.40e3T^{2} \) |
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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
−10.68232405634851902589570989711, −9.905230574243885891347888826347, −8.747086251027083076710653475898, −8.039693590171104259728137927811, −6.74046446217376670594265913957, −5.84748729181321712484739636398, −5.18239812997539733513693801004, −3.91138859619229722850555746975, −2.45050118720542817201807385452, −0.60216966320426496786487744722,
1.72138633999464382639174996234, 3.28935508887449376777213427644, 4.06577735785290869002778668212, 4.97447549134739976303217420557, 6.78085912956308853290755717978, 7.33501992065419931691419348182, 8.193992330621297062235446373865, 9.466759111526583418006468022658, 10.29558242420064949761129282774, 11.42223680124162766633302781280
