L(s) = 1 | + (−1.5 − 0.866i)3-s + (1.5 + 2.59i)9-s + 3·11-s − 2·13-s + 5.19i·17-s − 5.19i·19-s − 6·23-s − 5.19i·27-s + 10.3i·29-s + 3.46i·31-s + (−4.5 − 2.59i)33-s + 8·37-s + (3 + 1.73i)39-s + 5.19i·41-s + 3.46i·43-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (0.5 + 0.866i)9-s + 0.904·11-s − 0.554·13-s + 1.26i·17-s − 1.19i·19-s − 1.25·23-s − 0.999i·27-s + 1.92i·29-s + 0.622i·31-s + (−0.783 − 0.452i)33-s + 1.31·37-s + (0.480 + 0.277i)39-s + 0.811i·41-s + 0.528i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.095209602\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.095209602\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 - 5.19iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.976481671854491664169160539553, −8.966429718695992216810964432667, −8.120765976651224914736317122282, −7.11522854541445265490842928416, −6.55410115577573692881082713121, −5.69301132670595149418031678926, −4.76888859816786094262034067345, −3.82899820698980028168083646009, −2.30083713219875881569122642647, −1.10690793654110656542402701433,
0.63597144735035765118299971050, 2.29441613119590636210046084071, 3.86356699939317211641291694422, 4.37642736800182503572167199721, 5.60239882650989773031994577036, 6.11598497518027464190669059566, 7.13617777727614912949234822791, 7.940277479717745619180806268176, 9.181192802291979827371125520145, 9.754337560460811700069512717332