| L(s) = 1 | + (0.520 + 0.231i)2-s + (−1.62 + 0.597i)3-s + (−1.12 − 1.24i)4-s + (2.21 − 0.336i)5-s + (−0.984 − 0.0659i)6-s + (1.58 + 2.75i)7-s + (−0.646 − 1.99i)8-s + (2.28 − 1.94i)9-s + (1.22 + 0.336i)10-s + (1.73 + 0.772i)11-s + (2.56 + 1.35i)12-s + (5.02 − 2.23i)13-s + (0.189 + 1.79i)14-s + (−3.39 + 1.86i)15-s + (−0.225 + 2.14i)16-s + (0.888 + 2.73i)17-s + ⋯ |
| L(s) = 1 | + (0.367 + 0.163i)2-s + (−0.938 + 0.344i)3-s + (−0.560 − 0.622i)4-s + (0.988 − 0.150i)5-s + (−0.401 − 0.0269i)6-s + (0.600 + 1.04i)7-s + (−0.228 − 0.703i)8-s + (0.762 − 0.647i)9-s + (0.388 + 0.106i)10-s + (0.522 + 0.232i)11-s + (0.740 + 0.391i)12-s + (1.39 − 0.620i)13-s + (0.0505 + 0.481i)14-s + (−0.876 + 0.482i)15-s + (−0.0564 + 0.536i)16-s + (0.215 + 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25204 + 0.0969183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25204 + 0.0969183i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.62 - 0.597i)T \) |
| 5 | \( 1 + (-2.21 + 0.336i)T \) |
| good | 2 | \( 1 + (-0.520 - 0.231i)T + (1.33 + 1.48i)T^{2} \) |
| 7 | \( 1 + (-1.58 - 2.75i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.73 - 0.772i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-5.02 + 2.23i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.888 - 2.73i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.39 + 4.27i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.335 - 3.19i)T + (-22.4 + 4.78i)T^{2} \) |
| 29 | \( 1 + (1.75 - 0.373i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (7.35 + 1.56i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (4.57 + 3.32i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.974 - 0.433i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (3.64 + 6.31i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.34 + 1.56i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (0.982 - 3.02i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (12.0 - 5.37i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-0.958 - 0.426i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-12.7 - 2.70i)T + (61.2 + 27.2i)T^{2} \) |
| 71 | \( 1 + (2.10 - 6.47i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.173 - 0.126i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.79 + 1.01i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (10.0 - 11.1i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (2.15 - 1.56i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (13.1 - 2.80i)T + (88.6 - 39.4i)T^{2} \) |
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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
−12.42936911694346334652783170716, −11.16569431211523118887819322205, −10.45339209300123450145906801728, −9.316668161326972138681151945006, −8.750788992574316575244613690910, −6.68499757859594113039581892027, −5.60552499423282267606264464956, −5.42867793057593226971397096853, −3.97207795363037206962742928310, −1.50486351016520498090083715689,
1.52714267270283330888205751768, 3.73762530490596195729281441676, 4.83175221657701219981128042851, 5.97562350673640946340988760026, 6.99081139696808526970550767241, 8.195787867845575138787315908298, 9.367425557844263193112602085257, 10.63215259580686615269060020341, 11.24683070826857424988527378820, 12.29717532518352056734643434561
