| L(s) = 1 | + 5-s + 4·7-s − 4·13-s + 4·19-s + 23-s + 25-s − 6·29-s + 4·31-s + 4·35-s + 2·37-s + 4·43-s + 9·49-s − 6·53-s + 6·59-s + 8·61-s − 4·65-s + 4·67-s + 12·71-s − 10·73-s − 14·79-s − 6·83-s − 6·89-s − 16·91-s + 4·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.10·13-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.676·35-s + 0.328·37-s + 0.609·43-s + 9/7·49-s − 0.824·53-s + 0.781·59-s + 1.02·61-s − 0.496·65-s + 0.488·67-s + 1.42·71-s − 1.17·73-s − 1.57·79-s − 0.658·83-s − 0.635·89-s − 1.67·91-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.994976991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994976991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.77319574757562, −15.36499644877848, −14.61304168098264, −14.28683665183716, −13.99312286950404, −13.00721203043611, −12.75127861731405, −11.73529866826148, −11.58263786297998, −10.98299052478030, −10.17739352020043, −9.790598935777094, −9.081969403399109, −8.494085588758373, −7.744268545683415, −7.441036507264374, −6.715400353857339, −5.765596100322229, −5.284245327356352, −4.748802009652539, −4.112149380883046, −3.088110984948855, −2.298613373736647, −1.672730451992110, −0.7613971233175895,
0.7613971233175895, 1.672730451992110, 2.298613373736647, 3.088110984948855, 4.112149380883046, 4.748802009652539, 5.284245327356352, 5.765596100322229, 6.715400353857339, 7.441036507264374, 7.744268545683415, 8.494085588758373, 9.081969403399109, 9.790598935777094, 10.17739352020043, 10.98299052478030, 11.58263786297998, 11.73529866826148, 12.75127861731405, 13.00721203043611, 13.99312286950404, 14.28683665183716, 14.61304168098264, 15.36499644877848, 15.77319574757562
