| L(s) = 1 | + (−0.370 + 1.36i)2-s + (−0.365 − 1.69i)3-s + (−1.72 − 1.01i)4-s + (−1.04 − 0.138i)5-s + (2.44 + 0.127i)6-s + (−0.889 + 3.32i)7-s + (2.01 − 1.98i)8-s + (−2.73 + 1.23i)9-s + (0.577 − 1.38i)10-s + (−0.487 + 0.634i)11-s + (−1.08 + 3.29i)12-s + (0.231 − 0.177i)13-s + (−4.20 − 2.44i)14-s + (0.149 + 1.82i)15-s + (1.95 + 3.48i)16-s − 7.22·17-s + ⋯ |
| L(s) = 1 | + (−0.261 + 0.965i)2-s + (−0.211 − 0.977i)3-s + (−0.862 − 0.505i)4-s + (−0.469 − 0.0618i)5-s + (0.998 + 0.0521i)6-s + (−0.336 + 1.25i)7-s + (0.713 − 0.700i)8-s + (−0.910 + 0.412i)9-s + (0.182 − 0.437i)10-s + (−0.146 + 0.191i)11-s + (−0.311 + 0.950i)12-s + (0.0641 − 0.0492i)13-s + (−1.12 − 0.653i)14-s + (0.0387 + 0.472i)15-s + (0.489 + 0.872i)16-s − 1.75·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0189401 + 0.278919i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0189401 + 0.278919i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.370 - 1.36i)T \) |
| 3 | \( 1 + (0.365 + 1.69i)T \) |
| good | 5 | \( 1 + (1.04 + 0.138i)T + (4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.889 - 3.32i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.487 - 0.634i)T + (-2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.231 + 0.177i)T + (3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 7.22T + 17T^{2} \) |
| 19 | \( 1 + (2.42 - 5.85i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (5.46 - 1.46i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (0.246 + 1.87i)T + (-28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-5.11 - 2.95i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-10.4 + 4.33i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.976 + 3.64i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.309 - 0.237i)T + (11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-6.09 + 3.51i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.60 + 0.663i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-0.409 + 3.11i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (7.37 - 0.970i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (7.22 - 5.54i)T + (17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (11.2 - 11.2i)T - 71iT^{2} \) |
| 73 | \( 1 + (1.93 + 1.93i)T + 73iT^{2} \) |
| 79 | \( 1 + (2.05 + 3.55i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.664 + 5.04i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-6.64 - 6.64i)T + 89iT^{2} \) |
| 97 | \( 1 + (5.82 + 10.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.36255418954635957054931995844, −11.56813112723067700221552047741, −10.23678251255548013870695242471, −8.975648804166949795001694539902, −8.309101438222186384724794319214, −7.46197176284507661543344278890, −6.23648188560570679155571869832, −5.80602917536176237260882319775, −4.27316817052129498158966028350, −2.17577943718579870264807865895,
0.22400553547430313742401341392, 2.77536697844036039963727546525, 4.18909342721105742579546931774, 4.46398634976041392349089442878, 6.35849990888089552438820798752, 7.75777037685987664360357855003, 8.827875156514115952709627229214, 9.690023276481834776352552708810, 10.61103139765074687652872655340, 11.10625595182621472553429591807
