| L(s) = 1 | + (1.31 − 0.531i)2-s + (1.84 + 2.39i)3-s + (1.43 − 1.39i)4-s + (−2.83 − 2.17i)5-s + (3.68 + 2.16i)6-s + (2.04 + 1.68i)7-s + (1.14 − 2.58i)8-s + (−1.58 + 5.93i)9-s + (−4.87 − 1.34i)10-s + (−0.291 − 0.0383i)11-s + (5.98 + 0.878i)12-s + (−4.24 − 1.75i)13-s + (3.57 + 1.11i)14-s − 10.8i·15-s + (0.118 − 3.99i)16-s + (−2.00 − 1.15i)17-s + ⋯ |
| L(s) = 1 | + (0.926 − 0.375i)2-s + (1.06 + 1.38i)3-s + (0.717 − 0.696i)4-s + (−1.26 − 0.972i)5-s + (1.50 + 0.884i)6-s + (0.772 + 0.635i)7-s + (0.403 − 0.915i)8-s + (−0.529 + 1.97i)9-s + (−1.54 − 0.424i)10-s + (−0.0877 − 0.0115i)11-s + (1.72 + 0.253i)12-s + (−1.17 − 0.487i)13-s + (0.954 + 0.298i)14-s − 2.78i·15-s + (0.0297 − 0.999i)16-s + (−0.485 − 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.33020 + 0.225015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.33020 + 0.225015i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.31 + 0.531i)T \) |
| 7 | \( 1 + (-2.04 - 1.68i)T \) |
| good | 3 | \( 1 + (-1.84 - 2.39i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (2.83 + 2.17i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (0.291 + 0.0383i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (4.24 + 1.75i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (2.00 + 1.15i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.199 - 1.51i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (0.577 - 2.15i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.70 + 8.93i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (2.94 - 5.10i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.876 - 0.672i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (0.148 + 0.148i)T + 41iT^{2} \) |
| 43 | \( 1 + (0.752 + 1.81i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-0.734 + 0.423i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.07 - 0.273i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.834 + 6.33i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-4.14 + 0.545i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-5.77 - 7.53i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (10.2 - 10.2i)T - 71iT^{2} \) |
| 73 | \( 1 + (5.27 - 1.41i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.42 + 4.28i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.62 - 1.91i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.63 - 1.77i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−12.13469303586158412973133477846, −11.53820584878724966787753019713, −10.39697559430956914916429360347, −9.412546078511357265330833325724, −8.404178464033422742297020965206, −7.62062334914711593932077036907, −5.26662882640111314193044698993, −4.66376783626357991187227936236, −3.81208474664183179966554574781, −2.51427519431244916826836416160,
2.24388606265815779965765811644, 3.37200064356848490005756484486, 4.54783711189712299238313272280, 6.61358011504375642110143774596, 7.28716287280970207666786487168, 7.71267182109469379201900993775, 8.667021620874677304953245256853, 10.73799439867361230513242872166, 11.66523575263461635633703993808, 12.34578067799155357143755221170
