L(s) = 1 | − 0.915i·2-s + (0.5 + 0.866i)3-s + 1.16·4-s + (−2.26 + 1.30i)5-s + (0.792 − 0.457i)6-s + (2.57 − 0.608i)7-s − 2.89i·8-s + (−0.499 + 0.866i)9-s + (1.19 + 2.07i)10-s + (2.20 − 1.27i)11-s + (0.581 + 1.00i)12-s + (0.523 + 3.56i)13-s + (−0.557 − 2.35i)14-s + (−2.26 − 1.30i)15-s − 0.325·16-s + 6.19·17-s + ⋯ |
L(s) = 1 | − 0.647i·2-s + (0.288 + 0.499i)3-s + 0.581·4-s + (−1.01 + 0.585i)5-s + (0.323 − 0.186i)6-s + (0.973 − 0.230i)7-s − 1.02i·8-s + (−0.166 + 0.288i)9-s + (0.378 + 0.656i)10-s + (0.664 − 0.383i)11-s + (0.167 + 0.290i)12-s + (0.145 + 0.989i)13-s + (−0.148 − 0.629i)14-s + (−0.585 − 0.337i)15-s − 0.0813·16-s + 1.50·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55859 - 0.153532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55859 - 0.153532i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.57 + 0.608i)T \) |
| 13 | \( 1 + (-0.523 - 3.56i)T \) |
good | 2 | \( 1 + 0.915iT - 2T^{2} \) |
| 5 | \( 1 + (2.26 - 1.30i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.20 + 1.27i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 + (-2.36 - 1.36i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7.89T + 23T^{2} \) |
| 29 | \( 1 + (1.77 - 3.07i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.91 + 4.56i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.68iT - 37T^{2} \) |
| 41 | \( 1 + (3.98 + 2.30i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.11 + 3.65i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.53 - 0.887i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.42 - 5.92i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 2.25iT - 59T^{2} \) |
| 61 | \( 1 + (6.18 - 10.7i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.536 + 0.309i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.48 + 3.74i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (10.8 + 6.24i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.621 - 1.07i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.10iT - 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 + (1.32 - 0.767i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.68461509793670915159159482246, −11.09387944238182210771579547483, −10.26707124851208395539015920378, −9.175859886264470724594630605253, −7.82753623830523620790348483711, −7.27928137324140850852459053581, −5.78615303802889695375366839797, −4.00813167066585372883127006850, −3.51706300326752007841202511958, −1.77838662123423708571736291237,
1.58513769932380620795960385806, 3.40189063635528015235764934766, 4.95004069150170231447230815378, 5.97939620807102840891208288721, 7.34846044710861584225225335799, 7.963762954395169602736209647815, 8.453756318147344020378767495100, 9.993475134319351085021078507041, 11.40552297972545088067529771731, 11.89542485430465458999362802785