L(s) = 1 | + (1.58 − 0.916i)2-s + (0.678 − 1.17i)4-s − 0.645·5-s + 1.17i·8-s + (−1.02 + 0.591i)10-s + 5.31i·11-s + (−4.44 + 2.56i)13-s + (2.43 + 4.22i)16-s + (−0.814 − 1.41i)17-s + (2.09 + 1.20i)19-s + (−0.437 + 0.758i)20-s + (4.86 + 8.43i)22-s + 1.47i·23-s − 4.58·25-s + (−4.69 + 8.13i)26-s + ⋯ |
L(s) = 1 | + (1.12 − 0.647i)2-s + (0.339 − 0.587i)4-s − 0.288·5-s + 0.416i·8-s + (−0.323 + 0.187i)10-s + 1.60i·11-s + (−1.23 + 0.711i)13-s + (0.609 + 1.05i)16-s + (−0.197 − 0.342i)17-s + (0.479 + 0.276i)19-s + (−0.0978 + 0.169i)20-s + (1.03 + 1.79i)22-s + 0.306i·23-s − 0.916·25-s + (−0.921 + 1.59i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.205229479\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.205229479\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.58 + 0.916i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 0.645T + 5T^{2} \) |
| 11 | \( 1 - 5.31iT - 11T^{2} \) |
| 13 | \( 1 + (4.44 - 2.56i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.814 + 1.41i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.09 - 1.20i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.47iT - 23T^{2} \) |
| 29 | \( 1 + (-6.43 - 3.71i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.90 + 2.83i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.99 + 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.99 - 10.3i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.51 - 2.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.54 + 2.67i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.04 + 1.18i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.47 + 2.56i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.18 - 5.30i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.07 + 8.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (-10.2 + 5.90i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.48 + 6.02i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.51 + 6.09i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.16 - 3.74i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.3 - 8.31i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.764813838976989454320192330613, −9.267988534659213789227133917552, −7.86404509479246011778349722441, −7.35335954963619549618456896279, −6.30105374227306137794621739929, −5.09947302148130020370559209897, −4.61187662861249494428906349538, −3.82437888321356488602232603295, −2.63196690779138408841991751808, −1.84769301587960748345738648170,
0.61136957920628583842454825766, 2.71316804261990636781724552013, 3.56290387687441159699779621114, 4.49600606946065819879630362189, 5.40494154692435024639671947978, 5.96857873706415108610391055369, 6.88205239173233695126662980741, 7.71990249471267731130531992104, 8.446047729847793227300913303373, 9.524711116820059824801592423696