| L(s) = 1 | + (−0.137 − 0.163i)2-s + (−0.173 − 0.984i)3-s + (0.339 − 1.92i)4-s + (2.98 − 0.526i)5-s + (−0.137 + 0.163i)6-s + (−0.181 − 2.63i)7-s + (−0.730 + 0.421i)8-s + (−0.939 + 0.342i)9-s + (−0.494 − 0.415i)10-s + 1.47·11-s − 1.95·12-s + (3.17 + 2.66i)13-s + (−0.406 + 0.391i)14-s + (−1.03 − 2.84i)15-s + (−3.50 − 1.27i)16-s + (−1.26 + 3.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.0968 − 0.115i)2-s + (−0.100 − 0.568i)3-s + (0.169 − 0.962i)4-s + (1.33 − 0.235i)5-s + (−0.0559 + 0.0666i)6-s + (−0.0685 − 0.997i)7-s + (−0.258 + 0.149i)8-s + (−0.313 + 0.114i)9-s + (−0.156 − 0.131i)10-s + 0.444·11-s − 0.564·12-s + (0.881 + 0.739i)13-s + (−0.108 + 0.104i)14-s + (−0.267 − 0.735i)15-s + (−0.876 − 0.318i)16-s + (−0.306 + 0.842i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02127 - 1.20762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02127 - 1.20762i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.181 + 2.63i)T \) |
| 19 | \( 1 + (4.31 - 0.600i)T \) |
| good | 2 | \( 1 + (0.137 + 0.163i)T + (-0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-2.98 + 0.526i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + (-3.17 - 2.66i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.26 - 3.47i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.106 - 0.0895i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.71 - 0.654i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.616 - 1.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.18 - 2.41i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.94 + 5.83i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.43 + 0.524i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (3.68 + 10.1i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.21 - 0.743i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.23 - 1.90i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (5.19 - 6.18i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (6.08 - 7.25i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.60 - 7.14i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-12.6 + 2.23i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.251 + 0.691i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-13.1 - 7.58i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.16 - 6.58i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.19 - 12.4i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.70954956052831504865802582130, −10.33790581869454153719652630454, −9.272316175631745361715304618625, −8.489891124675155031229773362319, −6.79422921934895910890185975788, −6.40745917492430221277324085692, −5.46856987352083258275939071421, −4.11493021817986189448365267402, −2.07906239571368348727061662826, −1.22416094335848329698732517854,
2.28321431706907551733536934363, 3.25039267903640308127134446196, 4.73311467143999034561095261152, 5.99539185866190414656644106012, 6.51146611710868111262786960444, 8.042260742710383401087824927032, 8.962105372486124023582291924731, 9.489858200269964893839216679663, 10.63534400153192663886868756389, 11.44417926844787210162243333098
