L(s) = 1 | − 2.90·3-s + (−2.21 + 0.311i)5-s − 3.52i·7-s + 5.42·9-s + 3.80i·11-s + 2.62·13-s + (6.42 − 0.903i)15-s + 5.80i·17-s − 5.05i·19-s + 10.2i·21-s + 0.474i·23-s + (4.80 − 1.37i)25-s − 7.05·27-s + 2i·29-s − 2.75·31-s + ⋯ |
L(s) = 1 | − 1.67·3-s + (−0.990 + 0.139i)5-s − 1.33i·7-s + 1.80·9-s + 1.14i·11-s + 0.727·13-s + (1.65 − 0.233i)15-s + 1.40i·17-s − 1.15i·19-s + 2.23i·21-s + 0.0989i·23-s + (0.961 − 0.275i)25-s − 1.35·27-s + 0.371i·29-s − 0.494·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3312598261\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3312598261\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.21 - 0.311i)T \) |
good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 + 3.52iT - 7T^{2} \) |
| 11 | \( 1 - 3.80iT - 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 5.80iT - 17T^{2} \) |
| 19 | \( 1 + 5.05iT - 19T^{2} \) |
| 23 | \( 1 - 0.474iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 + 5.18T + 41T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 + 5.33iT - 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 + 5.05iT - 59T^{2} \) |
| 61 | \( 1 + 12.2iT - 61T^{2} \) |
| 67 | \( 1 + 7.76T + 67T^{2} \) |
| 71 | \( 1 - 4.85T + 71T^{2} \) |
| 73 | \( 1 + 6.66iT - 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−9.737741712956297337712430384872, −8.387381808916097415034534500188, −7.42317380216201800846709230739, −6.88761371469323909821033198649, −6.22285847895742343930524631207, −4.96275293020163801368425196248, −4.36704722236192362734842318888, −3.60998479776990115329557161954, −1.45294914689524267233933843123, −0.22594354613682051967255643017,
1.07546422311426779508781069030, 2.94210692351814796099076475427, 4.11055146910283247759856693277, 5.13077341445910684473441202879, 5.77915876504415876508341224264, 6.34587126817793682194049259692, 7.43119095880157122588968896722, 8.344293895602701374575988251743, 9.087862452015664506237653150114, 10.14232963033502136130391053739