| L(s) = 1 | + (1.88 + 0.779i)2-s + (−1.65 − 0.508i)3-s + (1.51 + 1.51i)4-s + (−2.03 + 0.405i)5-s + (−2.71 − 2.24i)6-s + (0.388 − 1.95i)7-s + (0.112 + 0.272i)8-s + (2.48 + 1.68i)9-s + (−4.14 − 0.825i)10-s + (1.30 + 1.95i)11-s + (−1.74 − 3.28i)12-s + (−4.24 + 4.24i)13-s + (2.25 − 3.37i)14-s + (3.57 + 0.364i)15-s − 3.68i·16-s + (4.01 + 0.958i)17-s + ⋯ |
| L(s) = 1 | + (1.33 + 0.550i)2-s + (−0.955 − 0.293i)3-s + (0.758 + 0.758i)4-s + (−0.911 + 0.181i)5-s + (−1.10 − 0.916i)6-s + (0.146 − 0.738i)7-s + (0.0399 + 0.0964i)8-s + (0.827 + 0.561i)9-s + (−1.31 − 0.260i)10-s + (0.394 + 0.590i)11-s + (−0.502 − 0.947i)12-s + (−1.17 + 1.17i)13-s + (0.602 − 0.901i)14-s + (0.924 + 0.0940i)15-s − 0.922i·16-s + (0.972 + 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06256 + 0.225038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06256 + 0.225038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 + 0.508i)T \) |
| 17 | \( 1 + (-4.01 - 0.958i)T \) |
| good | 2 | \( 1 + (-1.88 - 0.779i)T + (1.41 + 1.41i)T^{2} \) |
| 5 | \( 1 + (2.03 - 0.405i)T + (4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-0.388 + 1.95i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.30 - 1.95i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (4.24 - 4.24i)T - 13iT^{2} \) |
| 19 | \( 1 + (-1.93 + 4.67i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (0.734 - 0.491i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.0116 - 0.0587i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-2.44 - 1.63i)T + (11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (1.52 - 2.28i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (3.25 + 0.647i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (2.53 + 6.12i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (5.76 + 5.76i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.75 + 0.725i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-3.54 - 8.56i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-5.30 - 1.05i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 - 7.09iT - 67T^{2} \) |
| 71 | \( 1 + (-0.742 - 0.496i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (0.551 + 2.77i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (3.30 - 2.20i)T + (30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (1.44 - 3.49i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (2.00 - 2.00i)T - 89iT^{2} \) |
| 97 | \( 1 + (-18.4 + 3.67i)T + (89.6 - 37.1i)T^{2} \) |
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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
−15.47971983120873667496987877948, −14.46924597060954004240791090574, −13.41770568871911596832179593381, −12.09129199975039797870452170503, −11.68138390279922592156006398868, −9.978918889116017705364961727049, −7.37516199325656204483837660793, −6.84497158223642907508741513499, −5.07306446707463115808462440595, −4.06505897669567522333815926639,
3.44930311814817057569541000674, 4.93846067616521463250982613256, 5.89113617747745610555863645976, 7.962283304992587480691035312802, 10.01792845973655628299728605806, 11.43949544649856923149330613912, 12.07125346020183825739136996768, 12.70149683732359404323688483257, 14.40229287024635504915159506701, 15.26576160709783824117941630292
