| L(s) = 1 | + 2·13-s + 6·17-s + 4·19-s + 6·23-s − 6·29-s + 4·31-s − 2·37-s + 6·41-s + 10·43-s + 6·47-s − 6·53-s + 12·59-s − 2·61-s − 2·67-s + 12·71-s + 2·73-s + 8·79-s − 6·83-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s + 1.52·43-s + 0.875·47-s − 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.244·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s − 0.658·83-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.307569488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.307569488\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.60854895992694, −14.13553202257183, −13.75636077973061, −13.02158410809579, −12.67642573109773, −12.08668791301324, −11.56093277927159, −10.98942890112772, −10.61318429355468, −9.858266002301827, −9.428492608988158, −8.971962366843700, −8.253683703007742, −7.681877349377805, −7.297587982080817, −6.646114120208971, −5.838624716597478, −5.528856504674440, −4.906349726715783, −4.093601340408434, −3.501006815301042, −2.965553189500451, −2.196182336850910, −1.178246426729086, −0.7731435857637275,
0.7731435857637275, 1.178246426729086, 2.196182336850910, 2.965553189500451, 3.501006815301042, 4.093601340408434, 4.906349726715783, 5.528856504674440, 5.838624716597478, 6.646114120208971, 7.297587982080817, 7.681877349377805, 8.253683703007742, 8.971962366843700, 9.428492608988158, 9.858266002301827, 10.61318429355468, 10.98942890112772, 11.56093277927159, 12.08668791301324, 12.67642573109773, 13.02158410809579, 13.75636077973061, 14.13553202257183, 14.60854895992694
