L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.563 + 1.73i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (1.47 − 1.07i)6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.265 − 0.192i)9-s − 10-s + (−0.645 − 3.25i)11-s − 1.82·12-s + (1.79 + 1.30i)13-s + (0.309 − 0.951i)14-s + (0.563 + 1.73i)15-s + (−0.809 + 0.587i)16-s + (6.44 − 4.68i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.325 + 1.00i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (0.602 − 0.437i)6-s + (0.116 + 0.359i)7-s + (0.109 − 0.336i)8-s + (−0.0885 − 0.0643i)9-s − 0.316·10-s + (−0.194 − 0.980i)11-s − 0.526·12-s + (0.496 + 0.360i)13-s + (0.0825 − 0.254i)14-s + (0.145 + 0.447i)15-s + (−0.202 + 0.146i)16-s + (1.56 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15857 + 0.316755i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15857 + 0.316755i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.645 + 3.25i)T \) |
good | 3 | \( 1 + (0.563 - 1.73i)T + (-2.42 - 1.76i)T^{2} \) |
| 13 | \( 1 + (-1.79 - 1.30i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-6.44 + 4.68i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.282 - 0.870i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 6.07T + 23T^{2} \) |
| 29 | \( 1 + (-1.56 - 4.81i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (7.55 + 5.48i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.33 - 4.11i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.78 - 8.55i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 + (-1.30 + 4.02i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.55 - 5.48i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.22 - 6.84i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-10.2 + 7.45i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 + (0.0163 - 0.0118i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.66 - 11.2i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.59 - 5.52i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.604 - 0.439i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + (-5.88 - 4.27i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.31312011586934899753651209396, −9.580837494967958644493093114392, −8.991377381325425219725818860976, −8.095167785502507545922252123843, −7.02217102143773456888851705536, −5.65943097824290808444890743334, −5.10645257416206366315926812360, −3.81987274013234044346272611630, −2.85041684210329037609721673882, −1.15910193692239697504518769756,
1.00975620991228537046404803118, 2.04671297592137432021486251556, 3.70430618904254656635599658776, 5.25885798884002448217816745102, 6.03281047606277080026987110620, 7.01659972290362017822931910578, 7.42969089956977949549531332471, 8.328605352303578074139254696787, 9.402080240935582711198489334958, 10.28406121770160717925133708218