| L(s) = 1 | + (−0.943 − 1.05i)2-s + (2.23 − 1.12i)3-s + (−0.219 + 1.98i)4-s + (0.711 + 1.60i)5-s + (−3.29 − 1.29i)6-s + (2.06 + 1.23i)7-s + (2.30 − 1.64i)8-s + (1.93 − 2.61i)9-s + (1.01 − 2.26i)10-s + (1.25 + 0.977i)11-s + (1.74 + 4.68i)12-s + (−3.53 + 3.36i)13-s + (−0.643 − 3.34i)14-s + (3.39 + 2.78i)15-s + (−3.90 − 0.874i)16-s + (0.0832 + 0.101i)17-s + ⋯ |
| L(s) = 1 | + (−0.667 − 0.744i)2-s + (1.28 − 0.648i)3-s + (−0.109 + 0.993i)4-s + (0.318 + 0.717i)5-s + (−1.34 − 0.527i)6-s + (0.779 + 0.467i)7-s + (0.813 − 0.581i)8-s + (0.645 − 0.870i)9-s + (0.322 − 0.715i)10-s + (0.377 + 0.294i)11-s + (0.503 + 1.35i)12-s + (−0.980 + 0.933i)13-s + (−0.172 − 0.892i)14-s + (0.876 + 0.719i)15-s + (−0.975 − 0.218i)16-s + (0.0201 + 0.0246i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67690 - 0.435357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67690 - 0.435357i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.943 + 1.05i)T \) |
| good | 3 | \( 1 + (-2.23 + 1.12i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.711 - 1.60i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-2.06 - 1.23i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-1.25 - 0.977i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (3.53 - 3.36i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0832 - 0.101i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.171 + 0.989i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-2.95 - 1.39i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-5.01 - 2.84i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-3.34 + 0.666i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (8.21 + 5.20i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (7.29 + 8.05i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (3.07 + 9.31i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (3.49 + 1.86i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-6.46 + 3.66i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (3.92 - 4.11i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.0809 - 1.09i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-0.304 - 0.262i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-0.158 - 0.0234i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-1.60 + 0.959i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (13.2 + 4.01i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-5.18 + 3.28i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-2.90 + 1.37i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-2.73 - 1.83i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.67557841529025598727498133607, −9.855581162558265366775530644659, −8.815967955406247510716469059478, −8.536173627433886043131187072544, −7.19523357834878147202537065788, −6.96350382826932747285908207722, −4.88241040225941329097047788670, −3.47856884000489085807447575847, −2.39748459064449057285195960824, −1.80010153890736051961481355451,
1.33590928556745179438525061399, 2.94353814767323256694941288895, 4.56152449634421636989364099918, 5.11140858495002381995238681889, 6.58612138425887250413240013864, 7.84678871027056279430251805794, 8.281746310285337590854819398737, 9.032487530488738425349474337307, 9.883740836105316610376910793493, 10.39841968369132425590347301447
