L(s) = 1 | + i·2-s − 4-s + (2 + i)5-s − 2·7-s − i·8-s + (−1 + 2i)10-s − i·13-s − 2i·14-s + 16-s + (1 + 4i)17-s − 5·19-s + (−2 − i)20-s − 4·23-s + (3 + 4i)25-s + 26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.894 + 0.447i)5-s − 0.755·7-s − 0.353i·8-s + (−0.316 + 0.632i)10-s − 0.277i·13-s − 0.534i·14-s + 0.250·16-s + (0.242 + 0.970i)17-s − 1.14·19-s + (−0.447 − 0.223i)20-s − 0.834·23-s + (0.600 + 0.800i)25-s + 0.196·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.070533362\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.070533362\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 17 | \( 1 + (-1 - 4i)T \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 - 5iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 16iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−9.963123741944019862959305874844, −8.862049293104028361695830676164, −8.369545818840702615663150926401, −7.16329209308353280982807338089, −6.55740912371090858145672069243, −5.92822788596443623083495794559, −5.13308786733504418667309556553, −3.90337474231508462056542447153, −2.96980803656855517073432862747, −1.64649861219279977393751756354,
0.40114547560187481086930222676, 1.92951444925408139063775216266, 2.70980606848205391120750962180, 3.95107873855369251793007800520, 4.76213656868321918615353136769, 5.86511637571692962338817690437, 6.38551458610394572055480750248, 7.59335641965280810925057251078, 8.566728341050252038513865473742, 9.346163022835530793862130964887