| L(s) = 1 | + (−0.460 − 1.33i)2-s + (1.47 + 1.92i)3-s + (−1.57 + 1.23i)4-s + (0.169 + 0.130i)5-s + (1.88 − 2.85i)6-s + (−0.114 + 2.64i)7-s + (2.37 + 1.53i)8-s + (−0.740 + 2.76i)9-s + (0.0958 − 0.286i)10-s + (−5.24 − 0.690i)11-s + (−4.68 − 1.21i)12-s + (5.88 + 2.43i)13-s + (3.58 − 1.06i)14-s + 0.517i·15-s + (0.963 − 3.88i)16-s + (1.68 + 0.970i)17-s + ⋯ |
| L(s) = 1 | + (−0.325 − 0.945i)2-s + (0.850 + 1.10i)3-s + (−0.787 + 0.616i)4-s + (0.0758 + 0.0582i)5-s + (0.771 − 1.16i)6-s + (−0.0432 + 0.999i)7-s + (0.839 + 0.543i)8-s + (−0.246 + 0.921i)9-s + (0.0303 − 0.0907i)10-s + (−1.58 − 0.208i)11-s + (−1.35 − 0.349i)12-s + (1.63 + 0.676i)13-s + (0.958 − 0.284i)14-s + 0.133i·15-s + (0.240 − 0.970i)16-s + (0.407 + 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18071 + 0.346529i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18071 + 0.346529i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.460 + 1.33i)T \) |
| 7 | \( 1 + (0.114 - 2.64i)T \) |
| good | 3 | \( 1 + (-1.47 - 1.92i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (-0.169 - 0.130i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (5.24 + 0.690i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (-5.88 - 2.43i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (-1.68 - 0.970i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.124 - 0.946i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (0.264 - 0.987i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.48 + 6.00i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-2.26 + 3.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.98 - 5.35i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (8.40 + 8.40i)T + 41iT^{2} \) |
| 43 | \( 1 + (0.546 + 1.32i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (1.36 - 0.788i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.71 - 0.225i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.0306 + 0.232i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (5.31 - 0.699i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (0.0798 + 0.104i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-2.68 + 2.68i)T - 71iT^{2} \) |
| 73 | \( 1 + (-0.658 + 0.176i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (10.7 - 6.21i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.53 - 3.94i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-5.21 - 1.39i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−12.16350601657208358172712478829, −11.17158053747031941698353061777, −10.24340289946950452616048062887, −9.600180804748628171336341272923, −8.466544384424194383033823216315, −8.203930671172329989354303359996, −5.86589698855294229719141258830, −4.51766496212374003136212840866, −3.38590645321156826400685040873, −2.36026446583887874981625962582,
1.20624492573183654041113329811, 3.27359886158221473074025433976, 5.04627683826331851839678655293, 6.36449414119727604588521982919, 7.41073287756168300138031793163, 7.961870633507785476941638856773, 8.741397413239683242285830466652, 10.11581544531355262293765825658, 10.90248259188958432340775375989, 12.88624737963172619919250693400
