| L(s) = 1 | + (−1.93 + 0.340i)2-s + (−0.120 − 0.143i)3-s + (1.73 − 0.632i)4-s + (−1.25 − 1.85i)5-s + (0.282 + 0.236i)6-s + (−0.586 − 0.338i)7-s + (0.257 − 0.148i)8-s + (0.514 − 2.91i)9-s + (3.05 + 3.15i)10-s + (−1.42 − 2.46i)11-s + (−0.300 − 0.173i)12-s + (3.06 − 3.65i)13-s + (1.24 + 0.454i)14-s + (−0.115 + 0.403i)15-s + (−3.27 + 2.75i)16-s + (−5.10 + 0.900i)17-s + ⋯ |
| L(s) = 1 | + (−1.36 + 0.240i)2-s + (−0.0696 − 0.0830i)3-s + (0.868 − 0.316i)4-s + (−0.560 − 0.828i)5-s + (0.115 + 0.0966i)6-s + (−0.221 − 0.127i)7-s + (0.0909 − 0.0525i)8-s + (0.171 − 0.973i)9-s + (0.965 + 0.996i)10-s + (−0.428 − 0.741i)11-s + (−0.0867 − 0.0500i)12-s + (0.849 − 1.01i)13-s + (0.333 + 0.121i)14-s + (−0.0297 + 0.104i)15-s + (−0.819 + 0.687i)16-s + (−1.23 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.305746 - 0.262806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.305746 - 0.262806i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.25 + 1.85i)T \) |
| 19 | \( 1 + (-3.00 - 3.15i)T \) |
| good | 2 | \( 1 + (1.93 - 0.340i)T + (1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (0.120 + 0.143i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (0.586 + 0.338i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.42 + 2.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.06 + 3.65i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.10 - 0.900i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.359 - 0.987i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.247 + 1.40i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.135 - 0.234i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.603iT - 37T^{2} \) |
| 41 | \( 1 + (-5.15 + 4.32i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.92 + 5.28i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-7.77 - 1.37i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.35 - 6.47i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.75 - 9.96i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.02 + 2.55i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-4.05 - 0.714i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.14 - 2.60i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (10.3 + 12.3i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (11.3 - 9.54i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-12.2 - 7.09i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.31 + 5.29i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-5.64 + 0.994i)T + (91.1 - 33.1i)T^{2} \) |
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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
−13.55158880639905584538921093571, −12.66845674051122568521186069926, −11.36581494124719546248505679178, −10.31470003277994134029098444215, −9.073982916378652665559074771237, −8.422780580636817374432099760073, −7.34682637313588767967736201324, −5.87539506626635516050714128771, −3.85275884937924284790191142537, −0.77953019253088542684192252838,
2.34294012845249596644822820541, 4.55987093602384000263194400742, 6.77804835502070906318022166962, 7.66121257477730451149246856860, 8.836052741362452346320028077676, 9.915141440349968861715219284289, 10.97911178758635107640642644586, 11.43718208452034698848389893670, 13.18004448037835434152130654089, 14.25446748677594422824646632462
