| L(s) = 1 | + (−0.395 − 1.35i)2-s + (−0.654 − 0.755i)3-s + (−1.68 + 1.07i)4-s + (−0.236 − 1.64i)5-s + (−0.767 + 1.18i)6-s + (−1.60 + 3.51i)7-s + (2.12 + 1.86i)8-s + (−0.142 + 0.989i)9-s + (−2.14 + 0.972i)10-s + (1.51 − 5.16i)11-s + (1.91 + 0.571i)12-s + (−4.93 + 2.25i)13-s + (5.41 + 0.790i)14-s + (−1.08 + 1.25i)15-s + (1.69 − 3.62i)16-s + (−3.45 + 5.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.279 − 0.960i)2-s + (−0.378 − 0.436i)3-s + (−0.843 + 0.536i)4-s + (−0.105 − 0.736i)5-s + (−0.313 + 0.485i)6-s + (−0.607 + 1.32i)7-s + (0.751 + 0.659i)8-s + (−0.0474 + 0.329i)9-s + (−0.677 + 0.307i)10-s + (0.457 − 1.55i)11-s + (0.553 + 0.165i)12-s + (−1.36 + 0.625i)13-s + (1.44 + 0.211i)14-s + (−0.281 + 0.324i)15-s + (0.423 − 0.905i)16-s + (−0.836 + 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.541404 + 0.117370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.541404 + 0.117370i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.395 + 1.35i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-2.68 - 3.97i)T \) |
| good | 5 | \( 1 + (0.236 + 1.64i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.60 - 3.51i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.51 + 5.16i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (4.93 - 2.25i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (3.45 - 5.36i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.29 - 6.68i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.09 - 1.69i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.0568 + 0.0492i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.418 + 2.91i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.237 + 1.65i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-4.19 + 3.63i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 1.86iT - 47T^{2} \) |
| 53 | \( 1 + (2.25 - 4.93i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.71 - 5.93i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (9.64 - 11.1i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.551 - 1.87i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-0.0205 - 0.0698i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (12.0 - 7.74i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.56 - 7.80i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (9.65 + 1.38i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.591 + 0.512i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (10.0 - 1.44i)T + (93.0 - 27.3i)T^{2} \) |
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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
−11.03919798546758780746962787174, −9.941621607090063494121641740848, −8.924184837110962080810250640438, −8.689302981346831008776862178818, −7.46333978330778335419646463706, −5.98735629886669403570237542373, −5.34110631421832815928359049986, −3.95785189818834012694833353187, −2.74193496611087610263783446130, −1.42194223739343277059410618693,
0.39097981785538212610363812877, 2.97410910282124104279370307366, 4.58912182595891686605522288367, 4.84730860642751699025905379825, 6.63507913995398466069323937906, 7.06469086786669368530074869786, 7.52688145123806846619512934083, 9.334750811751732304564773531713, 9.677869836858165671198273819126, 10.46130719770872157368199663772
