| L(s) = 1 | + (1.57 − 1.81i)2-s + (−0.841 + 0.540i)3-s + (−0.533 − 3.71i)4-s + (−0.415 − 0.909i)5-s + (−0.341 + 2.37i)6-s + (1.31 − 0.386i)7-s + (−3.52 − 2.26i)8-s + (−0.830 + 1.81i)9-s + (−2.30 − 0.675i)10-s + (1.49 + 1.72i)11-s + (2.45 + 2.83i)12-s + (−0.817 − 0.239i)13-s + (1.36 − 2.99i)14-s + (0.841 + 0.540i)15-s + (−2.45 + 0.721i)16-s + (−0.581 + 4.04i)17-s + ⋯ |
| L(s) = 1 | + (1.11 − 1.28i)2-s + (−0.485 + 0.312i)3-s + (−0.266 − 1.85i)4-s + (−0.185 − 0.406i)5-s + (−0.139 + 0.968i)6-s + (0.497 − 0.146i)7-s + (−1.24 − 0.802i)8-s + (−0.276 + 0.606i)9-s + (−0.727 − 0.213i)10-s + (0.449 + 0.518i)11-s + (0.708 + 0.818i)12-s + (−0.226 − 0.0665i)13-s + (0.365 − 0.800i)14-s + (0.217 + 0.139i)15-s + (−0.614 + 0.180i)16-s + (−0.140 + 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0249 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0249 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05139 - 1.07801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05139 - 1.07801i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (1.93 + 4.38i)T \) |
| good | 2 | \( 1 + (-1.57 + 1.81i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (0.841 - 0.540i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (-1.31 + 0.386i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.49 - 1.72i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.817 + 0.239i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.581 - 4.04i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.384 - 2.67i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.951 - 6.61i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (8.68 + 5.57i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.36 + 9.56i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.86 - 6.27i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.91 + 3.79i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 8.71T + 47T^{2} \) |
| 53 | \( 1 + (6.55 - 1.92i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (2.60 + 0.765i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.764 - 0.491i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.49 + 7.49i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (1.29 - 1.49i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.15 - 8.03i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-14.1 - 4.14i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.727 - 1.59i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-4.47 + 2.87i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.52 - 9.91i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−12.89431622972217196854618167496, −12.39344710852233354954221673639, −11.20000919629303104246433429737, −10.75432402317271835089721458694, −9.534248819057749752228576962498, −7.913568982191460491132633761099, −5.89356701299972642021444507741, −4.82827373573867513169421665175, −3.92461660155874049383890990899, −1.95300775100368514876716889292,
3.42794605069091653498020812561, 4.92260337106200788416613313347, 6.01444942766143879275441166045, 6.90048651937873564396378165461, 7.88844753025572004448683415190, 9.286646718016693769531911006965, 11.29314980766677945548193340934, 11.90171036260673396936946140342, 13.08058594028714664500758583180, 14.06369397471529214128891989536
