L(s) = 1 | − i·3-s + 4i·5-s + 7-s − 9-s − 2i·11-s − 2i·13-s + 4·15-s + 4i·19-s − i·21-s − 6·23-s − 11·25-s + i·27-s − 10i·29-s + 8·31-s − 2·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.78i·5-s + 0.377·7-s − 0.333·9-s − 0.603i·11-s − 0.554i·13-s + 1.03·15-s + 0.917i·19-s − 0.218i·21-s − 1.25·23-s − 2.20·25-s + 0.192i·27-s − 1.85i·29-s + 1.43·31-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.651391457\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.651391457\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 4iT - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.914145707921738133403406725023, −7.46622627239294114428881909080, −6.62104368656006805370353397972, −6.00946840182085616718758070914, −5.60098784489637143028928377903, −4.11199312657434901007562626676, −3.53886973196677936682033838393, −2.54167632811218658911028489417, −2.07270132819158300984593415062, −0.50038391571746192260768362228,
0.975264575845687092298604501138, 1.81038538175177717885896848489, 2.97704023972658309508803840147, 4.29389922522866444379896838667, 4.51083648083749805569463861392, 5.12251949578498814796842571696, 5.92042196353171964992490414435, 6.82393730542878431093505433913, 7.80679283497321633511723011342, 8.415614398720588469531414886863