| L(s) = 1 | + (0.548 − 2.40i)2-s + (−3.32 + 4.16i)3-s + (1.73 + 0.835i)4-s + (9.40 − 11.7i)5-s + (8.18 + 10.2i)6-s + (12.9 − 13.2i)7-s + (15.2 − 19.1i)8-s + (−0.303 − 1.33i)9-s + (−23.1 − 29.0i)10-s + (−11.8 + 51.7i)11-s + (−9.24 + 4.45i)12-s + (11.7 − 51.3i)13-s + (−24.8 − 38.2i)14-s + (17.8 + 78.2i)15-s + (−27.9 − 35.0i)16-s + (−106. + 51.2i)17-s + ⋯ |
| L(s) = 1 | + (0.193 − 0.849i)2-s + (−0.639 + 0.801i)3-s + (0.216 + 0.104i)4-s + (0.840 − 1.05i)5-s + (0.556 + 0.698i)6-s + (0.696 − 0.717i)7-s + (0.674 − 0.845i)8-s + (−0.0112 − 0.0492i)9-s + (−0.732 − 0.918i)10-s + (−0.324 + 1.41i)11-s + (−0.222 + 0.107i)12-s + (0.249 − 1.09i)13-s + (−0.474 − 0.730i)14-s + (0.307 + 1.34i)15-s + (−0.437 − 0.548i)16-s + (−1.51 + 0.731i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.46038 - 0.649121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46038 - 0.649121i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-12.9 + 13.2i)T \) |
| good | 2 | \( 1 + (-0.548 + 2.40i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (3.32 - 4.16i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-9.40 + 11.7i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (11.8 - 51.7i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-11.7 + 51.3i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (106. - 51.2i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 43.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-97.1 - 46.7i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (218. - 105. i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 171.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (30.5 - 14.7i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-119. + 150. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-191. - 239. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (73.0 - 319. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (40.5 + 19.5i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-185. - 232. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-22.2 + 10.7i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 269.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (663. + 319. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-19.9 - 87.5i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 162.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (65.3 + 286. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-61.1 - 267. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 505.T + 9.12e5T^{2} \) |
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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
−15.14056418872877926130372174934, −13.21686289197974986517352633902, −12.78137679163615977203538548036, −11.14293733438683023828883084068, −10.52972014131105981353905135025, −9.410874433353763666225205319126, −7.47259069621776645715029755068, −5.29974406958337607031447516052, −4.29725492548471713743754634163, −1.70763359310699212531912885351,
2.16481855343906270035138184990, 5.51047773748220430345714556069, 6.35649021178457007708675028736, 7.22312309108438656579720409574, 8.964052784491981511014093558041, 11.14954712983071849577763916495, 11.34986494685058681716735438696, 13.36443600216424909025239476934, 14.18135538742158009464800763155, 15.17579791415236207504130424137
