| L(s) = 1 | + (0.445 − 1.94i)2-s + (−4.69 + 5.89i)3-s + (−3.60 − 1.73i)4-s + (9.89 − 12.4i)5-s + (9.39 + 11.7i)6-s + (−8.41 + 16.4i)7-s + (−4.98 + 6.25i)8-s + (−6.62 − 29.0i)9-s + (−19.7 − 24.8i)10-s + (−13.6 + 59.8i)11-s + (27.1 − 13.0i)12-s + (−15.5 + 68.1i)13-s + (28.4 + 23.7i)14-s + (26.6 + 116. i)15-s + (9.97 + 12.5i)16-s + (20.3 − 9.77i)17-s + ⋯ |
| L(s) = 1 | + (0.157 − 0.689i)2-s + (−0.904 + 1.13i)3-s + (−0.450 − 0.216i)4-s + (0.885 − 1.10i)5-s + (0.639 + 0.801i)6-s + (−0.454 + 0.890i)7-s + (−0.220 + 0.276i)8-s + (−0.245 − 1.07i)9-s + (−0.625 − 0.784i)10-s + (−0.374 + 1.64i)11-s + (0.653 − 0.314i)12-s + (−0.332 + 1.45i)13-s + (0.542 + 0.453i)14-s + (0.458 + 2.00i)15-s + (0.155 + 0.195i)16-s + (0.289 − 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0137 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0137 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.646627 + 0.637770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.646627 + 0.637770i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.445 + 1.94i)T \) |
| 7 | \( 1 + (8.41 - 16.4i)T \) |
| good | 3 | \( 1 + (4.69 - 5.89i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-9.89 + 12.4i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (13.6 - 59.8i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (15.5 - 68.1i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-20.3 + 9.77i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-58.6 - 28.2i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-185. + 89.3i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 - 75.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + (39.9 - 19.2i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (262. - 329. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (47.1 + 59.1i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-17.9 + 78.7i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-559. - 269. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (391. + 491. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (259. - 125. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 314.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (508. + 244. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-97.5 - 427. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 724.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (168. + 739. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (114. + 501. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−13.35694591313367941178505022545, −12.36602158957264280366758010186, −11.76114206454633136484177520322, −10.22653748888518212095750490804, −9.664118903192675867402253189692, −8.876510663771336847344209886919, −6.33379125495113145135553773052, −5.01581028270718872129713830924, −4.52073060647597968629406627318, −2.06155558513072014046921027508,
0.54212465057352033700298978749, 3.06176265670349641991406689314, 5.55637526714801712522675381040, 6.33690745597401900130368253326, 7.06958446407744909978229471672, 8.299232146844717421248408404879, 10.35442865480317457847350246992, 10.80679187154242672584352651759, 12.45155240392442188532503822788, 13.37543588268853704941609779101
