| L(s) = 1 | + (−0.0687 + 0.917i)2-s + (−1.23 + 0.844i)3-s + (1.14 + 0.171i)4-s + (0.162 − 0.712i)5-s + (−0.689 − 1.19i)6-s + (−2.59 − 0.535i)7-s + (−0.645 + 2.82i)8-s + (−0.274 + 0.699i)9-s + (0.642 + 0.198i)10-s + (−1.56 + 3.99i)11-s + (−1.55 + 0.750i)12-s + (1.79 + 1.66i)13-s + (0.669 − 2.34i)14-s + (0.400 + 1.02i)15-s + (−0.345 − 0.106i)16-s + (−0.0710 − 0.311i)17-s + ⋯ |
| L(s) = 1 | + (−0.0486 + 0.648i)2-s + (−0.715 + 0.487i)3-s + (0.570 + 0.0859i)4-s + (0.0727 − 0.318i)5-s + (−0.281 − 0.487i)6-s + (−0.979 − 0.202i)7-s + (−0.228 + 1.00i)8-s + (−0.0914 + 0.233i)9-s + (0.203 + 0.0626i)10-s + (−0.473 + 1.20i)11-s + (−0.449 + 0.216i)12-s + (0.496 + 0.461i)13-s + (0.178 − 0.625i)14-s + (0.103 + 0.263i)15-s + (−0.0864 − 0.0266i)16-s + (−0.0172 − 0.0754i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.219385 + 0.847408i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.219385 + 0.847408i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.59 + 0.535i)T \) |
| 43 | \( 1 + (-5.20 + 3.98i)T \) |
| good | 2 | \( 1 + (0.0687 - 0.917i)T + (-1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (1.23 - 0.844i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.162 + 0.712i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (1.56 - 3.99i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.79 - 1.66i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.0710 + 0.311i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (2.49 - 3.12i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (3.75 + 4.70i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (0.663 - 8.85i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (-0.532 + 7.10i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 - 0.728T + 37T^{2} \) |
| 41 | \( 1 + (-3.92 + 1.88i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-2.61 - 6.67i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-1.49 - 0.461i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-8.83 + 8.20i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.534 - 7.13i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (2.62 + 6.68i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (-3.04 - 7.75i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (-3.38 + 14.8i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (-5.64 - 9.77i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.16 - 15.5i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (2.31 + 1.11i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (6.27 + 7.86i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−12.23265027203097655115781032362, −10.97445490162627347117339700829, −10.43331473271057846018315007593, −9.362163684070651568444793981236, −8.161145896086455092906984779270, −7.09778516344593644788570265240, −6.24326648213119944458452806660, −5.34188463370720073860278319151, −4.15571765374865680037339730203, −2.34583841982055695242346442973,
0.67836607375149406921372870922, 2.64481404509484187093012072128, 3.59056513280606571898310047942, 5.79943057474235479123980461201, 6.23375112588568044831709374547, 7.17284525701271497361703657109, 8.594189434765981832443481444265, 9.778400065428122319279444040685, 10.69310598040620939542979878674, 11.31878961780591917139649374402
