L(s) = 1 | − 0.792i·2-s − i·3-s + 1.37·4-s + i·5-s − 0.792·6-s + (−0.792 + 2.52i)7-s − 2.67i·8-s − 9-s + 0.792·10-s + 3.31i·11-s − 1.37i·12-s + (2 + 0.627i)14-s + 15-s + 0.627·16-s + 4.10·17-s + 0.792i·18-s + ⋯ |
L(s) = 1 | − 0.560i·2-s − 0.577i·3-s + 0.686·4-s + 0.447i·5-s − 0.323·6-s + (−0.299 + 0.954i)7-s − 0.944i·8-s − 0.333·9-s + 0.250·10-s + 1.00i·11-s − 0.396i·12-s + (0.534 + 0.167i)14-s + 0.258·15-s + 0.156·16-s + 0.996·17-s + 0.186i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.959788746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.959788746\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.792 - 2.52i)T \) |
| 11 | \( 1 - 3.31iT \) |
good | 2 | \( 1 + 0.792iT - 2T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.10T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 - 9.11T + 23T^{2} \) |
| 29 | \( 1 - 5.98iT - 29T^{2} \) |
| 31 | \( 1 + 6.74iT - 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 4.10iT - 43T^{2} \) |
| 47 | \( 1 - 12.7iT - 47T^{2} \) |
| 53 | \( 1 + 1.62T + 53T^{2} \) |
| 59 | \( 1 - 1.62iT - 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 4.74T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 0.294iT - 79T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + 8.37iT - 89T^{2} \) |
| 97 | \( 1 - 8.37iT - 97T^{2} \) |
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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
−9.689353894208461392183008383826, −9.225949501494232240575865377819, −7.894374904957657000767079683714, −7.27963153165073579265763950493, −6.41063525545116294148447969656, −5.79383626589021402094273522979, −4.44120281988975934742965116865, −2.99806500440245206801233574842, −2.52786415122708487258114392561, −1.33851787351539540582707758707,
0.957187527536910719350204287617, 2.72623034965580240768414812452, 3.68933993402173404153773691524, 4.76872191063629861759604731761, 5.70975649348881083359645252525, 6.44149241458695279424714464050, 7.36279788456547766003765403082, 8.128099424767217500183714952439, 8.906422994103390523326914921816, 9.846573118627587886167112385170