| L(s) = 1 | + (−2.24 − 0.999i)2-s + (0.330 − 3.14i)3-s + (2.69 + 2.99i)4-s + 2.52·5-s + (−3.88 + 6.73i)6-s + (2.19 + 2.44i)7-s + (−1.54 − 4.75i)8-s + (−6.87 − 1.46i)9-s + (−5.66 − 2.52i)10-s + (3.84 − 0.816i)11-s + (10.3 − 7.51i)12-s + (−3.53 + 0.724i)13-s + (−2.49 − 7.67i)14-s + (0.835 − 7.94i)15-s + (−0.440 + 4.18i)16-s + (5.56 + 1.18i)17-s + ⋯ |
| L(s) = 1 | + (−1.58 − 0.706i)2-s + (0.191 − 1.81i)3-s + (1.34 + 1.49i)4-s + 1.12·5-s + (−1.58 + 2.75i)6-s + (0.830 + 0.922i)7-s + (−0.546 − 1.68i)8-s + (−2.29 − 0.487i)9-s + (−1.79 − 0.797i)10-s + (1.15 − 0.246i)11-s + (2.98 − 2.16i)12-s + (−0.979 + 0.200i)13-s + (−0.666 − 2.05i)14-s + (0.215 − 2.05i)15-s + (−0.110 + 1.04i)16-s + (1.35 + 0.286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.324123 - 0.838941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.324123 - 0.838941i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (3.53 - 0.724i)T \) |
| 31 | \( 1 + (0.816 + 5.50i)T \) |
| good | 2 | \( 1 + (2.24 + 0.999i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.330 + 3.14i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 + (-2.19 - 2.44i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-3.84 + 0.816i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (-5.56 - 1.18i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.647 + 6.16i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-2.09 + 2.32i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (4.89 + 2.18i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (1.06 + 1.84i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.14 - 1.84i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.898 - 8.54i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (1.19 + 0.866i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.444 + 1.36i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.04 - 0.465i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (3.47 - 6.01i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.51 - 4.34i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.62 - 1.19i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (0.0104 - 0.0323i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-3.47 - 10.6i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (5.20 - 3.77i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-10.0 + 2.13i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (0.433 + 0.480i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−11.13049508126983179171173696381, −9.597110090601574708018017667980, −9.138520826848659634632892957648, −8.251994151533440997709774873428, −7.48979991774472071199733039601, −6.56634119368611664568768927973, −5.56567233548867764793627848978, −2.66658962027152238329004001463, −2.03571629030074365566418429696, −1.08951008203564505810011693682,
1.62751561005442055460845114328, 3.67178379113185527878415045661, 5.06242647196057369871971353798, 5.86464560933513796521665478057, 7.25955128590130412801290952086, 8.158097788285003585327497665391, 9.319951435623231468724105864314, 9.566130322581641521958937677858, 10.36626453995034196874021512821, 10.76591247241159277873638907741
