| L(s) = 1 | + (−0.205 − 0.109i)3-s + (−1.29 + 1.58i)5-s + (−1.60 − 1.06i)7-s + (−1.63 − 2.44i)9-s + (2.28 + 0.693i)11-s + (−1.01 + 0.828i)13-s + (0.440 − 0.182i)15-s + (−6.19 − 2.56i)17-s + (0.717 − 7.28i)19-s + (0.211 + 0.395i)21-s + (−6.82 − 1.35i)23-s + (0.159 + 0.804i)25-s + (0.135 + 1.37i)27-s + (−1.11 − 3.69i)29-s + (0.0726 − 0.0726i)31-s + ⋯ |
| L(s) = 1 | + (−0.118 − 0.0634i)3-s + (−0.580 + 0.706i)5-s + (−0.604 − 0.404i)7-s + (−0.545 − 0.816i)9-s + (0.688 + 0.208i)11-s + (−0.280 + 0.229i)13-s + (0.113 − 0.0471i)15-s + (−1.50 − 0.621i)17-s + (0.164 − 1.67i)19-s + (0.0461 + 0.0863i)21-s + (−1.42 − 0.282i)23-s + (0.0319 + 0.160i)25-s + (0.0261 + 0.265i)27-s + (−0.207 − 0.685i)29-s + (0.0130 − 0.0130i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.132444 - 0.363638i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132444 - 0.363638i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (0.205 + 0.109i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (1.29 - 1.58i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (1.60 + 1.06i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-2.28 - 0.693i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (1.01 - 0.828i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (6.19 + 2.56i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.717 + 7.28i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (6.82 + 1.35i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (1.11 + 3.69i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-0.0726 + 0.0726i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.87 + 0.282i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.658 + 3.30i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (3.06 - 1.63i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (1.75 - 4.23i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (2.91 - 9.59i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (4.93 + 4.04i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (0.733 - 1.37i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (2.54 - 4.75i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (1.60 - 2.39i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-11.8 + 7.92i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (3.13 + 7.56i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-10.8 - 1.06i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-12.0 + 2.38i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (2.03 - 2.03i)T - 97iT^{2} \) |
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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
−10.73098515109383020676521811136, −9.487967179596626182499370948948, −9.027023745724379500614513602122, −7.64035835911573045748320029218, −6.74234118991406906338885748820, −6.31107640613943074378693480356, −4.63205629204703846914705225090, −3.67050300997332689102608199139, −2.55368318982984299885783932522, −0.21944606174914145246371095050,
1.97415340837877228261246848457, 3.56083814374621746612907495986, 4.54720501564157527416382898106, 5.70512609206626077395782265040, 6.50306996803602312269332994756, 7.966258040236638330912048931924, 8.418769534832008349393093981938, 9.417963467756951123344993466434, 10.36542155276541601329042483270, 11.33800905000769582871796306158
