L(s) = 1 | + (1.39 + 0.221i)2-s + 2.04i·3-s + (1.90 + 0.619i)4-s + (0.665 + 2.13i)5-s + (−0.453 + 2.85i)6-s − 2.07·7-s + (2.51 + 1.28i)8-s − 1.17·9-s + (0.456 + 3.12i)10-s − 4.08i·11-s + (−1.26 + 3.88i)12-s − 4.52·13-s + (−2.90 − 0.460i)14-s + (−4.36 + 1.36i)15-s + (3.23 + 2.35i)16-s − 6.58i·17-s + ⋯ |
L(s) = 1 | + (0.987 + 0.156i)2-s + 1.17i·3-s + (0.950 + 0.309i)4-s + (0.297 + 0.954i)5-s + (−0.184 + 1.16i)6-s − 0.785·7-s + (0.890 + 0.454i)8-s − 0.392·9-s + (0.144 + 0.989i)10-s − 1.23i·11-s + (−0.365 + 1.12i)12-s − 1.25·13-s + (−0.775 − 0.123i)14-s + (−1.12 + 0.351i)15-s + (0.808 + 0.588i)16-s − 1.59i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59511 + 1.80495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59511 + 1.80495i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.221i)T \) |
| 5 | \( 1 + (-0.665 - 2.13i)T \) |
| 19 | \( 1 + (-4.31 + 0.590i)T \) |
good | 3 | \( 1 - 2.04iT - 3T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 + 4.08iT - 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 + 6.58iT - 17T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 - 5.90iT - 29T^{2} \) |
| 31 | \( 1 + 1.69T + 31T^{2} \) |
| 37 | \( 1 - 1.81T + 37T^{2} \) |
| 41 | \( 1 + 1.10iT - 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 + 9.98T + 47T^{2} \) |
| 53 | \( 1 - 2.18T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 - 4.67T + 71T^{2} \) |
| 73 | \( 1 + 6.18iT - 73T^{2} \) |
| 79 | \( 1 + 4.04T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 + 0.553iT - 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−11.35094171985234816732235633687, −10.90393409808437946362447381513, −9.847436776590652125558722795787, −9.234643419957944452638224261855, −7.42733623794248805744611104272, −6.78207938053724380337583779102, −5.52857319977800698221260333936, −4.80753075507507316549961053901, −3.23357381077206660638385525297, −2.99963726510501941773024337974,
1.41070139079176588871796032322, 2.52798093092974735794584871864, 4.19992400427360114672645526581, 5.24927074353721677306433379299, 6.30456340304103716379632337932, 7.15433034962940062767474912856, 7.915333997832085495640214377120, 9.512142083323084776354187744416, 10.13230547172991460490909031034, 11.64302552723702052770799780411