L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.07 − 1.96i)5-s + (−2.59 + 0.507i)7-s − 0.999·8-s + (−1.16 − 1.90i)10-s + (−4.30 − 2.48i)11-s + (−1.15 − 1.99i)13-s + (−0.858 + 2.50i)14-s + (−0.5 + 0.866i)16-s − 4.05i·17-s + 1.63i·19-s + (−2.23 + 0.0519i)20-s + (−4.30 + 2.48i)22-s + (3.70 + 6.41i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.479 − 0.877i)5-s + (−0.981 + 0.191i)7-s − 0.353·8-s + (−0.367 − 0.603i)10-s + (−1.29 − 0.749i)11-s + (−0.319 − 0.553i)13-s + (−0.229 + 0.668i)14-s + (−0.125 + 0.216i)16-s − 0.984i·17-s + 0.375i·19-s + (−0.499 + 0.0116i)20-s + (−0.917 + 0.529i)22-s + (0.772 + 1.33i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3880504809\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3880504809\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.07 + 1.96i)T \) |
| 7 | \( 1 + (2.59 - 0.507i)T \) |
good | 11 | \( 1 + (4.30 + 2.48i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.15 + 1.99i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.05iT - 17T^{2} \) |
| 19 | \( 1 - 1.63iT - 19T^{2} \) |
| 23 | \( 1 + (-3.70 - 6.41i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.86 - 3.96i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.00 - 3.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 10.4iT - 37T^{2} \) |
| 41 | \( 1 + (-2.40 - 4.17i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.31 + 1.33i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.03 + 5.21i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + (5.82 + 10.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.89 - 2.24i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.266 - 0.153i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.38iT - 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + (1.42 - 2.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.60 + 5.54i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 + (-4.81 + 8.34i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.788567884398723860783575996549, −8.142035566357621831376737472196, −7.03160373732404905939391578790, −6.01334144958768571451744715760, −5.24492841751716435493778051887, −4.84599113672074006101811102099, −3.22031401522750409105608112870, −2.93471553784546632851061691296, −1.44595393730662450562177609462, −0.11910837072381254332320889120,
2.25224846085527967332571001130, 2.94617140532152190055653712846, 4.06193710134786564721337286657, 4.95522246473590407599430913049, 5.99096004531187872632093475632, 6.53206793694194723638855717372, 7.25311136308107594861229739093, 7.895597872311094883735054729065, 9.039586572554670371070525063439, 9.705983445435284253178534549982