| L(s) = 1 | + (0.131 + 0.746i)2-s + (0.766 − 0.642i)3-s + (1.33 − 0.487i)4-s + (−2.97 − 1.08i)5-s + (0.580 + 0.487i)6-s + (−1.89 + 3.28i)7-s + (1.29 + 2.24i)8-s + (0.173 − 0.984i)9-s + (0.416 − 2.36i)10-s + (−0.752 − 1.30i)11-s + (0.712 − 1.23i)12-s + (−2.32 − 1.95i)13-s + (−2.70 − 0.984i)14-s + (−2.97 + 1.08i)15-s + (0.674 − 0.566i)16-s + (0.685 + 3.88i)17-s + ⋯ |
| L(s) = 1 | + (0.0931 + 0.528i)2-s + (0.442 − 0.371i)3-s + (0.669 − 0.243i)4-s + (−1.32 − 0.483i)5-s + (0.237 + 0.198i)6-s + (−0.717 + 1.24i)7-s + (0.459 + 0.795i)8-s + (0.0578 − 0.328i)9-s + (0.131 − 0.746i)10-s + (−0.226 − 0.393i)11-s + (0.205 − 0.356i)12-s + (−0.646 − 0.542i)13-s + (−0.722 − 0.263i)14-s + (−0.767 + 0.279i)15-s + (0.168 − 0.141i)16-s + (0.166 + 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.939110 + 0.122790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.939110 + 0.122790i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-1.20 + 4.18i)T \) |
| good | 2 | \( 1 + (-0.131 - 0.746i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (2.97 + 1.08i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.89 - 3.28i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.752 + 1.30i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.32 + 1.95i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.685 - 3.88i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-6.29 + 2.29i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.277 - 1.57i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.46 - 4.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.28T + 37T^{2} \) |
| 41 | \( 1 + (0.0274 - 0.0230i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.21 - 2.99i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.05 + 5.99i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (7.88 - 2.86i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.07 + 6.09i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (0.317 - 0.115i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.08 + 6.16i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.75 - 1.72i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (2.53 - 2.13i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (1.87 - 1.57i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (4.27 - 7.40i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.1 + 9.36i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.00 - 17.0i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−15.49510102177538067051579637043, −14.60338709497921701148372833351, −12.81506979916143387223938191730, −12.13604822963849891314651194777, −10.89780693262650979596136769846, −8.978132508365731257460324157678, −7.989682697023698520744181531858, −6.81243635672524381051641350086, −5.29089623669116790937221565825, −2.96494445695469741312837951016,
3.14669803106301511008458342044, 4.15844514488188378657574386378, 7.14626900670603290352186575343, 7.54006170257855262897383577799, 9.675476662149904235434394124935, 10.71827446902335487305913753369, 11.65735892120152933352156862762, 12.77135860580238214455966907632, 14.11597998661181739679167349909, 15.34395318339125923915458148073
