| 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
−14.06369397471529214128891989536, −13.08058594028714664500758583180, −11.90171036260673396936946140342, −11.29314980766677945548193340934, −9.286646718016693769531911006965, −7.88844753025572004448683415190, −6.90048651937873564396378165461, −6.01444942766143879275441166045, −4.92260337106200788416613313347, −3.42794605069091653498020812561,
1.95300775100368514876716889292, 3.92461660155874049383890990899, 4.82827373573867513169421665175, 5.89356701299972642021444507741, 7.913568982191460491132633761099, 9.534248819057749752228576962498, 10.75432402317271835089721458694, 11.20000919629303104246433429737, 12.39344710852233354954221673639, 12.89431622972217196854618167496
