L(s) = 1 | + (1.32 + 0.5i)2-s − 2.64·3-s + (1.50 + 1.32i)4-s + (−3.50 − 1.32i)6-s + 4i·7-s + (1.32 + 2.50i)8-s + 4.00·9-s + 2.64i·11-s + (−3.96 − 3.50i)12-s + (−2 + 5.29i)14-s + (0.500 + 3.96i)16-s − 3i·17-s + (5.29 + 2.00i)18-s − 2.64i·19-s − 10.5i·21-s + (−1.32 + 3.50i)22-s + ⋯ |
L(s) = 1 | + (0.935 + 0.353i)2-s − 1.52·3-s + (0.750 + 0.661i)4-s + (−1.42 − 0.540i)6-s + 1.51i·7-s + (0.467 + 0.883i)8-s + 1.33·9-s + 0.797i·11-s + (−1.14 − 1.01i)12-s + (−0.534 + 1.41i)14-s + (0.125 + 0.992i)16-s − 0.727i·17-s + (1.24 + 0.471i)18-s − 0.606i·19-s − 2.30i·21-s + (−0.282 + 0.746i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.877212 + 0.897666i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.877212 + 0.897666i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.32 - 0.5i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 2.64iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 2.64iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 7.93T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−12.45104116409666147860285301126, −11.87949871979181209258594158038, −11.30954030282382953610889116730, −10.03201445799250182539782768531, −8.586212018657907730774990811163, −7.08350979943628006797820629194, −6.22374337506708269031862472825, −5.33017780950534025516329791163, −4.59533903115521122854606799298, −2.51054797241481095901075845225,
1.06737227812559465230353803346, 3.65581506697939024858887814472, 4.68010370972612999007388483110, 5.85502028149049305838723519637, 6.58208622324050151875550052676, 7.73565745543270063342162743222, 9.938037413481134233377458492389, 10.65539273333467234358171186297, 11.27225617672738947921449054437, 12.08864456148825753122715781053