L(s) = 1 | + (−0.980 − 0.195i)2-s + (0.923 + 0.382i)4-s + (0.831 − 0.555i)7-s + (−0.831 − 0.555i)8-s + (−0.555 + 0.831i)9-s + (0.577 + 1.90i)11-s + (−0.923 + 0.382i)14-s + (0.707 + 0.707i)16-s + (0.707 − 0.707i)18-s + (−0.195 − 1.98i)22-s + (−1.08 + 0.216i)23-s + (0.195 − 0.980i)25-s + (0.980 − 0.195i)28-s + (1.47 + 0.448i)29-s + (−0.555 − 0.831i)32-s + ⋯ |
L(s) = 1 | + (−0.980 − 0.195i)2-s + (0.923 + 0.382i)4-s + (0.831 − 0.555i)7-s + (−0.831 − 0.555i)8-s + (−0.555 + 0.831i)9-s + (0.577 + 1.90i)11-s + (−0.923 + 0.382i)14-s + (0.707 + 0.707i)16-s + (0.707 − 0.707i)18-s + (−0.195 − 1.98i)22-s + (−1.08 + 0.216i)23-s + (0.195 − 0.980i)25-s + (0.980 − 0.195i)28-s + (1.47 + 0.448i)29-s + (−0.555 − 0.831i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6978523514\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6978523514\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.980 + 0.195i)T \) |
| 7 | \( 1 + (-0.831 + 0.555i)T \) |
good | 3 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 5 | \( 1 + (-0.195 + 0.980i)T^{2} \) |
| 11 | \( 1 + (-0.577 - 1.90i)T + (-0.831 + 0.555i)T^{2} \) |
| 13 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 19 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 23 | \( 1 + (1.08 - 0.216i)T + (0.923 - 0.382i)T^{2} \) |
| 29 | \( 1 + (-1.47 - 0.448i)T + (0.831 + 0.555i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-0.0569 + 0.577i)T + (-0.980 - 0.195i)T^{2} \) |
| 41 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.902 - 1.68i)T + (-0.555 - 0.831i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 53 | \( 1 + (-1.68 + 0.512i)T + (0.831 - 0.555i)T^{2} \) |
| 59 | \( 1 + (0.195 - 0.980i)T^{2} \) |
| 61 | \( 1 + (-0.555 + 0.831i)T^{2} \) |
| 67 | \( 1 + (-0.831 + 0.444i)T + (0.555 - 0.831i)T^{2} \) |
| 71 | \( 1 + (0.425 + 0.636i)T + (-0.382 + 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (-0.149 + 0.360i)T + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 89 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.24103743255152468981431781299, −9.742516766053554616544143623636, −8.577159219521256182759346641821, −7.965365630906974042733495291504, −7.22845173347030464321691076977, −6.40109625679992229575998062523, −4.98454304857285747195775485911, −4.11459427672052244550643508241, −2.48999041705503352304867473775, −1.59719120759213681589720411310,
1.06235904301495902466601847447, 2.59719565561349382622181878261, 3.68674637273595465041086824178, 5.41171355618386908578883569194, 6.02738631215242691339858494558, 6.83401013490854556919713358548, 8.160951228310254993712647373874, 8.554542405464162037797023769561, 9.135895195307424229118327673176, 10.21447024458927994343672933187