L(s) = 1 | + (1.48 + 1.77i)2-s + (−2.36 − 1.98i)3-s + (−0.585 + 3.32i)4-s + (−0.253 − 0.696i)5-s − 7.15i·6-s + (−1.02 + 0.373i)7-s + (−2.75 + 1.58i)8-s + (1.13 + 6.41i)9-s + (0.858 − 1.48i)10-s + (−0.376 − 0.651i)11-s + (7.96 − 6.68i)12-s + (2.68 + 0.472i)13-s + (−2.19 − 1.26i)14-s + (−0.782 + 2.14i)15-s + (−0.586 − 0.213i)16-s + (−5.77 + 1.01i)17-s + ⋯ |
L(s) = 1 | + (1.05 + 1.25i)2-s + (−1.36 − 1.14i)3-s + (−0.292 + 1.66i)4-s + (−0.113 − 0.311i)5-s − 2.91i·6-s + (−0.388 + 0.141i)7-s + (−0.973 + 0.561i)8-s + (0.377 + 2.13i)9-s + (0.271 − 0.470i)10-s + (−0.113 − 0.196i)11-s + (2.30 − 1.93i)12-s + (0.743 + 0.131i)13-s + (−0.586 − 0.338i)14-s + (−0.201 + 0.554i)15-s + (−0.146 − 0.0533i)16-s + (−1.40 + 0.247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.773449 + 0.311352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.773449 + 0.311352i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (0.224 + 6.07i)T \) |
good | 2 | \( 1 + (-1.48 - 1.77i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (2.36 + 1.98i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (0.253 + 0.696i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.02 - 0.373i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.376 + 0.651i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.68 - 0.472i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (5.77 - 1.01i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-1.47 + 1.75i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (3.96 + 2.29i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.18 + 1.83i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.48iT - 31T^{2} \) |
| 41 | \( 1 + (0.929 - 5.27i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 - 3.73iT - 43T^{2} \) |
| 47 | \( 1 + (-2.26 + 3.91i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (11.8 + 4.33i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.26 - 3.48i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (2.14 + 0.378i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.49 + 1.63i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.02 - 1.69i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + (-2.84 - 7.80i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.488 - 2.77i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.96 + 13.6i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.861 - 0.497i)T + (48.5 + 84.0i)T^{2} \) |
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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
−16.32518373073747834536146287227, −15.80104284086235224362893380055, −13.98403636739718407927280433941, −13.04761963305510953796900582387, −12.34288132240599741844350075402, −11.00281253916290118313931333075, −8.251888870440711251234983925193, −6.77215989279496157814366133766, −6.12866928915493983104911318246, −4.74655846318169567945592336148,
3.65646125861576805955934975133, 4.88774580614743777362381093074, 6.24879925168430773268026007002, 9.632111816225522775860906252949, 10.69444977905674560003087960958, 11.32220021480129665835581897232, 12.37052175753133555992614686589, 13.64700230229218723914466858809, 15.18204215899467971166486359973, 16.05734013410283849834033949912