| L(s) = 1 | + (−1.26 + 0.537i)2-s + (−1.15 − 1.29i)3-s + (−0.0684 + 0.0705i)4-s + (−2.16 + 1.53i)5-s + (2.16 + 1.01i)6-s + (2.38 − 0.924i)7-s + (1.04 − 2.69i)8-s + (−0.333 + 2.98i)9-s + (1.92 − 3.10i)10-s + (0.605 − 0.801i)11-s + (0.170 + 0.00684i)12-s + (−0.228 + 1.47i)13-s + (−2.53 + 2.45i)14-s + (4.47 + 1.02i)15-s + (0.116 + 3.79i)16-s + (−6.02 − 2.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.897 + 0.380i)2-s + (−0.666 − 0.745i)3-s + (−0.0342 + 0.0352i)4-s + (−0.967 + 0.685i)5-s + (0.881 + 0.415i)6-s + (0.902 − 0.349i)7-s + (0.369 − 0.953i)8-s + (−0.111 + 0.993i)9-s + (0.608 − 0.983i)10-s + (0.182 − 0.241i)11-s + (0.0491 + 0.00197i)12-s + (−0.0633 + 0.407i)13-s + (−0.677 + 0.656i)14-s + (1.15 + 0.264i)15-s + (0.0292 + 0.947i)16-s + (−1.46 − 0.514i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.452976 + 0.164486i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.452976 + 0.164486i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.15 + 1.29i)T \) |
| 103 | \( 1 + (6.49 + 7.80i)T \) |
| good | 2 | \( 1 + (1.26 - 0.537i)T + (1.39 - 1.43i)T^{2} \) |
| 5 | \( 1 + (2.16 - 1.53i)T + (1.66 - 4.71i)T^{2} \) |
| 7 | \( 1 + (-2.38 + 0.924i)T + (5.17 - 4.71i)T^{2} \) |
| 11 | \( 1 + (-0.605 + 0.801i)T + (-3.01 - 10.5i)T^{2} \) |
| 13 | \( 1 + (0.228 - 1.47i)T + (-12.3 - 3.94i)T^{2} \) |
| 17 | \( 1 + (6.02 + 2.12i)T + (13.2 + 10.6i)T^{2} \) |
| 19 | \( 1 + (3.68 - 0.227i)T + (18.8 - 2.33i)T^{2} \) |
| 23 | \( 1 + (0.964 + 7.78i)T + (-22.3 + 5.61i)T^{2} \) |
| 29 | \( 1 + (1.60 - 3.48i)T + (-18.8 - 22.0i)T^{2} \) |
| 31 | \( 1 + (-10.2 + 0.314i)T + (30.9 - 1.90i)T^{2} \) |
| 37 | \( 1 + (2.14 + 2.35i)T + (-3.41 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-8.02 - 0.743i)T + (40.3 + 7.53i)T^{2} \) |
| 43 | \( 1 + (-1.43 - 4.50i)T + (-35.0 + 24.8i)T^{2} \) |
| 47 | \( 1 + 7.54T + 47T^{2} \) |
| 53 | \( 1 + (-3.00 - 4.52i)T + (-20.6 + 48.8i)T^{2} \) |
| 59 | \( 1 + (0.461 - 1.19i)T + (-43.6 - 39.7i)T^{2} \) |
| 61 | \( 1 + (-8.17 - 9.54i)T + (-9.35 + 60.2i)T^{2} \) |
| 67 | \( 1 + (-14.6 + 2.28i)T + (63.8 - 20.3i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 7.82i)T + (23.5 + 66.9i)T^{2} \) |
| 73 | \( 1 + (7.42 + 0.688i)T + (71.7 + 13.4i)T^{2} \) |
| 79 | \( 1 + (-12.0 + 8.53i)T + (26.2 - 74.5i)T^{2} \) |
| 83 | \( 1 + (-14.7 - 2.29i)T + (79.0 + 25.1i)T^{2} \) |
| 89 | \( 1 + (0.973 - 3.42i)T + (-75.6 - 46.8i)T^{2} \) |
| 97 | \( 1 + (0.965 - 1.12i)T + (-14.8 - 95.8i)T^{2} \) |
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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
−10.38032721344922312035537419763, −9.019870835874154886356922561349, −8.213321195598743699562554082675, −7.77082866010914280579687627070, −6.75212904140349922602749660090, −6.54490397349636323100015608285, −4.72404899733299383785932085797, −4.12267257317694319242859863515, −2.38707332500997306917552479668, −0.75740677548403099554304513387,
0.55257793560275834902682946464, 2.01561681269896977965500950422, 3.92207906373326511534206336790, 4.67430204532822190793286591487, 5.32084222431358656095823855891, 6.53435023010137819230964331523, 8.033192372731241250007621954420, 8.344425819065336665174825552456, 9.223628276185851796447746242750, 9.930548097735078518340348516642
