| L(s) = 1 | + (0.422 + 0.487i)2-s + (−1.61 + 0.621i)3-s + (0.225 − 1.56i)4-s + (−1.17 − 0.344i)5-s + (−0.986 − 0.526i)6-s + (2.44 − 2.11i)7-s + (1.94 − 1.25i)8-s + (2.22 − 2.00i)9-s + (−0.328 − 0.718i)10-s + (0.398 + 0.116i)11-s + (0.608 + 2.67i)12-s + (0.261 − 0.406i)13-s + (2.06 + 0.297i)14-s + (2.11 − 0.171i)15-s + (−1.60 − 0.471i)16-s + (3.72 − 0.535i)17-s + ⋯ |
| L(s) = 1 | + (0.298 + 0.345i)2-s + (−0.933 + 0.358i)3-s + (0.112 − 0.783i)4-s + (−0.524 − 0.154i)5-s + (−0.402 − 0.214i)6-s + (0.924 − 0.801i)7-s + (0.688 − 0.442i)8-s + (0.742 − 0.669i)9-s + (−0.103 − 0.227i)10-s + (0.120 + 0.0352i)11-s + (0.175 + 0.771i)12-s + (0.0724 − 0.112i)13-s + (0.552 + 0.0794i)14-s + (0.545 − 0.0443i)15-s + (−0.401 − 0.117i)16-s + (0.903 − 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04027 - 0.323650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04027 - 0.323650i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.61 - 0.621i)T \) |
| 67 | \( 1 + (2.78 - 7.69i)T \) |
| good | 2 | \( 1 + (-0.422 - 0.487i)T + (-0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (1.17 + 0.344i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-2.44 + 2.11i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.398 - 0.116i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.261 + 0.406i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.72 + 0.535i)T + (16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.610 + 0.704i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (5.29 + 2.41i)T + (15.0 + 17.3i)T^{2} \) |
| 29 | \( 1 - 3.04iT - 29T^{2} \) |
| 31 | \( 1 + (-3.37 - 5.25i)T + (-12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 - 9.29T + 37T^{2} \) |
| 41 | \( 1 + (-0.592 - 4.11i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.27 - 0.327i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (8.54 + 3.90i)T + (30.7 + 35.5i)T^{2} \) |
| 53 | \( 1 + (0.172 - 1.19i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-6.53 - 10.1i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.63 - 8.99i)T + (-51.3 + 32.9i)T^{2} \) |
| 71 | \( 1 + (1.95 + 0.281i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-9.73 + 2.85i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.40 + 5.29i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (4.08 - 13.9i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-13.0 + 5.96i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 - 2.68iT - 97T^{2} \) |
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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
−12.10931878676686528598448623320, −11.36335481736679545772475102433, −10.46562123145711599834446932175, −9.802723648178393387744135349766, −8.088799533618746755764514779538, −7.05246147667980850292140864378, −5.94787440311798198413387501990, −4.87803102131149468350848801678, −4.10108381047321876248558566472, −1.11175365362279698411639729381,
2.02692242665691410323071892339, 3.83564607829388678814912674902, 5.03277388530964713001853163285, 6.20145720596186186448357102072, 7.75965032421152893012884066362, 8.032446202766538511348667206195, 9.794397198558091403230020798567, 11.24349924134824412433314341118, 11.60396756833814826177777072840, 12.23415030893856872855052635308
