| L(s) = 1 | + (−1.37 + 0.333i)2-s + (0.755 − 0.654i)3-s + (1.77 − 0.916i)4-s + (1.90 − 0.273i)5-s + (−0.820 + 1.15i)6-s + (1.21 − 2.65i)7-s + (−2.13 + 1.85i)8-s + (0.142 − 0.989i)9-s + (−2.52 + 1.01i)10-s + (0.357 − 1.21i)11-s + (0.743 − 1.85i)12-s + (1.67 − 0.765i)13-s + (−0.780 + 4.05i)14-s + (1.26 − 1.45i)15-s + (2.32 − 3.25i)16-s + (1.80 + 1.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.971 + 0.235i)2-s + (0.436 − 0.378i)3-s + (0.888 − 0.458i)4-s + (0.852 − 0.122i)5-s + (−0.334 + 0.470i)6-s + (0.458 − 1.00i)7-s + (−0.755 + 0.654i)8-s + (0.0474 − 0.329i)9-s + (−0.799 + 0.319i)10-s + (0.107 − 0.366i)11-s + (0.214 − 0.535i)12-s + (0.465 − 0.212i)13-s + (−0.208 + 1.08i)14-s + (0.325 − 0.375i)15-s + (0.580 − 0.814i)16-s + (0.437 + 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16916 - 0.608964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16916 - 0.608964i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.333i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (2.90 - 3.81i)T \) |
| good | 5 | \( 1 + (-1.90 + 0.273i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.21 + 2.65i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.357 + 1.21i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 0.765i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.80 - 1.16i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (2.43 + 3.78i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.814 - 1.26i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (2.29 - 2.64i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.0379 + 0.00545i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.229 - 1.59i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-9.53 + 8.26i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 + (-2.76 - 1.26i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-6.29 + 2.87i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.68 - 3.19i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (3.58 + 12.1i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.13 - 0.625i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-1.10 + 0.709i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-6.01 - 13.1i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-9.71 - 1.39i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.34 - 7.31i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.798 + 5.55i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.55361617592146646017683248409, −9.643826595594738107143650069647, −8.882014517048715850570033506297, −8.004481845010229509263988927440, −7.24998086226664225676058853891, −6.31111207869099653001719358527, −5.38036587305454122095130125131, −3.69292552255475784531587661220, −2.16803973533880990956642984459, −1.06569183858588918591613338447,
1.80670050895178846932316516860, 2.57483382941559467563724924430, 4.03667536434326821532929319844, 5.61776523890513259670104373867, 6.35874302515137393442146447624, 7.70096522985784247661250784428, 8.464946772863009362374810314939, 9.234620468709277553655082673803, 9.899651345377063023912488561173, 10.63687948025792749800137761478
