| L(s) = 1 | + (−0.166 − 1.16i)2-s + (−0.526 + 1.15i)3-s + (0.598 − 0.175i)4-s + (−0.654 + 0.755i)5-s + (1.42 + 0.419i)6-s + (4.23 − 2.72i)7-s + (−1.27 − 2.79i)8-s + (0.911 + 1.05i)9-s + (0.986 + 0.634i)10-s + (−0.353 + 2.45i)11-s + (−0.112 + 0.783i)12-s + (−2.10 − 1.35i)13-s + (−3.86 − 4.46i)14-s + (−0.526 − 1.15i)15-s + (−1.98 + 1.27i)16-s + (−1.81 − 0.531i)17-s + ⋯ |
| L(s) = 1 | + (−0.118 − 0.821i)2-s + (−0.304 + 0.666i)3-s + (0.299 − 0.0879i)4-s + (−0.292 + 0.337i)5-s + (0.582 + 0.171i)6-s + (1.59 − 1.02i)7-s + (−0.452 − 0.989i)8-s + (0.303 + 0.350i)9-s + (0.312 + 0.200i)10-s + (−0.106 + 0.740i)11-s + (−0.0325 + 0.226i)12-s + (−0.583 − 0.375i)13-s + (−1.03 − 1.19i)14-s + (−0.136 − 0.297i)15-s + (−0.496 + 0.319i)16-s + (−0.439 − 0.128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00901 - 0.346085i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00901 - 0.346085i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (0.932 - 4.70i)T \) |
| good | 2 | \( 1 + (0.166 + 1.16i)T + (-1.91 + 0.563i)T^{2} \) |
| 3 | \( 1 + (0.526 - 1.15i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (-4.23 + 2.72i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.353 - 2.45i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.10 + 1.35i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.81 + 0.531i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (3.66 - 1.07i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.95 - 0.866i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.365 + 0.800i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (3.98 + 4.60i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (0.518 - 0.598i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (4.06 - 8.90i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 + (-11.8 + 7.59i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (9.56 + 6.14i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.20 - 7.02i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.625 - 4.34i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.25 + 8.74i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-5.14 + 1.51i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-11.0 - 7.10i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.60 - 3.00i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.784 + 1.71i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (3.27 - 3.78i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−13.33221193981482490549310487593, −12.02913259898926141082119947276, −11.10386568571637851553875577991, −10.58370692186741881835289564828, −9.799031702886855033532691084428, −7.931801796664949994645019383925, −7.01572315793790611183216462081, −4.99296379584500863140995636813, −3.98075114116852031835654305014, −1.90721035349816323932420210009,
2.11676736085133563669090584411, 4.78394154561024648374268459313, 5.98179253329769002885300126279, 7.05640959038236505846326550971, 8.239764918561527887437836289900, 8.757151800445721027265176236254, 10.88727754034003444194575709630, 11.86311524966447967943050971099, 12.30673338973438599856719663996, 13.88679818924456218232010564966
