| L(s) = 1 | + (1.35 + 2.11i)3-s + (3.81 − 1.74i)5-s + (−1.23 + 0.361i)7-s + (−1.37 + 3.00i)9-s + (2.10 − 1.82i)11-s + (−0.900 + 3.06i)13-s + (8.86 + 5.69i)15-s + (−0.200 + 1.39i)17-s + (−5.93 + 0.853i)19-s + (−2.43 − 2.11i)21-s + (4.77 + 0.469i)23-s + (8.26 − 9.53i)25-s + (−0.764 + 0.109i)27-s + (5.36 + 0.771i)29-s + (−2.59 − 1.66i)31-s + ⋯ |
| L(s) = 1 | + (0.783 + 1.21i)3-s + (1.70 − 0.779i)5-s + (−0.465 + 0.136i)7-s + (−0.457 + 1.00i)9-s + (0.634 − 0.549i)11-s + (−0.249 + 0.850i)13-s + (2.29 + 1.47i)15-s + (−0.0486 + 0.338i)17-s + (−1.36 + 0.195i)19-s + (−0.531 − 0.460i)21-s + (0.995 + 0.0978i)23-s + (1.65 − 1.90i)25-s + (−0.147 + 0.0211i)27-s + (0.995 + 0.143i)29-s + (−0.465 − 0.299i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.29526 + 0.998595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29526 + 0.998595i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-4.77 - 0.469i)T \) |
| good | 3 | \( 1 + (-1.35 - 2.11i)T + (-1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (-3.81 + 1.74i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (1.23 - 0.361i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.10 + 1.82i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.900 - 3.06i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.200 - 1.39i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (5.93 - 0.853i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-5.36 - 0.771i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (2.59 + 1.66i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (6.97 + 3.18i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.478 - 1.04i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (0.886 + 1.37i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 + (3.24 + 11.0i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (2.45 - 8.35i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (4.47 - 6.95i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (6.96 + 6.03i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-3.19 + 3.68i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.59 - 11.1i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (3.30 + 0.971i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (6.86 + 3.13i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-4.04 + 2.59i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (2.85 + 6.24i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.17398894751025915059686621189, −9.565490864623702439235544456367, −8.798141812600068724345695945233, −8.689128883431813772726151066100, −6.73188680300736796504942577860, −5.97365370873320272783435883533, −4.93839392723977909753846297335, −4.11593113796346456038957997721, −2.88050692145918180699751223732, −1.72597609059588517821518449784,
1.46754379987880506753618907345, 2.43195208168598898256057481306, 3.15154613135780572969992875754, 5.01032100426979091776215828509, 6.32296386804609988358000096558, 6.66115885439397686348499745880, 7.44471554260090300100500284728, 8.647812346919408941080956757334, 9.341830285553363954622329909333, 10.21029648958247634822172058733
