| L(s) = 1 | + (−0.764 − 2.35i)2-s + (−0.574 − 0.186i)3-s + (−1.71 + 1.24i)4-s + 2.23·5-s + 1.49i·6-s + (−5.48 + 3.98i)7-s + (−3.76 − 2.73i)8-s + (−6.98 − 5.07i)9-s + (−1.70 − 5.26i)10-s + (−12.0 − 16.5i)11-s + (1.21 − 0.395i)12-s + (9.12 + 2.96i)13-s + (13.5 + 9.86i)14-s + (−1.28 − 0.417i)15-s + (−6.17 + 19.0i)16-s + (−10.5 + 14.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 − 1.17i)2-s + (−0.191 − 0.0622i)3-s + (−0.428 + 0.311i)4-s + 0.447·5-s + 0.249i·6-s + (−0.783 + 0.569i)7-s + (−0.470 − 0.341i)8-s + (−0.776 − 0.563i)9-s + (−0.170 − 0.526i)10-s + (−1.09 − 1.50i)11-s + (0.101 − 0.0329i)12-s + (0.701 + 0.228i)13-s + (0.969 + 0.704i)14-s + (−0.0856 − 0.0278i)15-s + (−0.385 + 1.18i)16-s + (−0.618 + 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.144554 + 0.557291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144554 + 0.557291i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 2.23T \) |
| 31 | \( 1 + (2.11 + 30.9i)T \) |
| good | 2 | \( 1 + (0.764 + 2.35i)T + (-3.23 + 2.35i)T^{2} \) |
| 3 | \( 1 + (0.574 + 0.186i)T + (7.28 + 5.29i)T^{2} \) |
| 7 | \( 1 + (5.48 - 3.98i)T + (15.1 - 46.6i)T^{2} \) |
| 11 | \( 1 + (12.0 + 16.5i)T + (-37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-9.12 - 2.96i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (10.5 - 14.4i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (3.01 + 9.27i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-6.77 + 9.32i)T + (-163. - 503. i)T^{2} \) |
| 29 | \( 1 + (-3.41 + 1.11i)T + (680. - 494. i)T^{2} \) |
| 37 | \( 1 - 36.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (19.7 + 60.8i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (-48.4 + 15.7i)T + (1.49e3 - 1.08e3i)T^{2} \) |
| 47 | \( 1 + (-18.0 + 55.5i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-7.45 + 10.2i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-12.6 + 38.8i)T + (-2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 - 23.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 127.T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-91.7 - 66.6i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (69.0 + 95.0i)T + (-1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-70.5 + 97.1i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (11.0 - 3.60i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (23.7 + 32.6i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (2.05 - 1.49i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−11.93667114663343880712970727341, −11.02323691695690260266160381558, −10.40206435984623646942037570842, −9.049016255360028172784803988543, −8.604377705913326390717374414683, −6.36934309727996135295426753806, −5.77613535272490558624904281716, −3.46819949943082352162799118017, −2.47758365892057707322312939367, −0.39005904130831276431335687009,
2.71219944284364950559891291786, 4.89172365233430884116571601029, 5.94863787714638014732847488710, 7.02278228702671616537558176810, 7.84559774760785126774351236770, 9.062735893120527640792321322995, 10.07399095296219583282134522500, 11.08214181051768533712021331725, 12.51373900419776758140460795031, 13.45763758474202566971219237346
