| L(s) = 1 | + (−1.14 − 1.36i)2-s + (0.815 − 2.24i)3-s + (−0.205 + 1.16i)4-s + (−2.08 − 0.816i)5-s + (−3.99 + 1.45i)6-s + (3.67 + 2.11i)7-s + (−1.25 + 0.727i)8-s + (−2.05 − 1.72i)9-s + (1.27 + 3.78i)10-s + (0.245 + 0.425i)11-s + (2.44 + 1.41i)12-s + (−1.42 − 3.91i)13-s + (−1.31 − 7.45i)14-s + (−3.52 + 3.99i)15-s + (4.66 + 1.69i)16-s + (1.30 + 1.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.811 − 0.966i)2-s + (0.470 − 1.29i)3-s + (−0.102 + 0.583i)4-s + (−0.930 − 0.365i)5-s + (−1.63 + 0.594i)6-s + (1.38 + 0.801i)7-s + (−0.445 + 0.257i)8-s + (−0.684 − 0.574i)9-s + (0.402 + 1.19i)10-s + (0.0740 + 0.128i)11-s + (0.706 + 0.407i)12-s + (−0.394 − 1.08i)13-s + (−0.351 − 1.99i)14-s + (−0.910 + 1.03i)15-s + (1.16 + 0.424i)16-s + (0.317 + 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.209421 - 0.704739i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.209421 - 0.704739i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.08 + 0.816i)T \) |
| 19 | \( 1 + (1.86 + 3.93i)T \) |
| good | 2 | \( 1 + (1.14 + 1.36i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.815 + 2.24i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-3.67 - 2.11i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.245 - 0.425i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.42 + 3.91i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.30 - 1.55i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-4.32 - 0.763i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.49 - 2.09i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.04 - 3.54i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.14iT - 37T^{2} \) |
| 41 | \( 1 + (4.10 + 1.49i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (10.4 - 1.84i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.68 + 2.00i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-11.2 - 1.98i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.415 + 0.348i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.36 - 13.4i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.60 - 5.48i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.04 - 5.94i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.702 + 1.93i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-5.01 - 1.82i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (7.02 + 4.05i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.07 - 1.48i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (3.62 + 4.32i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−13.11927658014480161446513524270, −12.20880673535627324542391991658, −11.63591171371241529261662251707, −10.54714034759382061549477238113, −8.692946150707048772822892001030, −8.373938072269828801755401307390, −7.26757427379445087211358827791, −5.19120471502499298996963505107, −2.76662487457768720806940891167, −1.31744481535857768329780072088,
3.70823474995577619206244853078, 4.78256268097420870078995642286, 6.92324969624131911172863965719, 7.920053166969581283137820618568, 8.713822638903871065664489022900, 9.889635130800325604120171435180, 10.92704938439449172946325733496, 11.95045375526882795454974332513, 14.13347221729045938675476846618, 14.80867129509478956789043884455
