| L(s) = 1 | − 2·5-s − 4·11-s + 6·13-s − 4·17-s − 4·19-s − 25-s + 10·29-s − 2·31-s − 37-s − 2·41-s − 4·43-s − 6·53-s + 8·55-s + 14·61-s − 12·65-s + 12·67-s + 8·71-s + 14·73-s + 12·79-s + 6·83-s + 8·85-s + 8·95-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.20·11-s + 1.66·13-s − 0.970·17-s − 0.917·19-s − 1/5·25-s + 1.85·29-s − 0.359·31-s − 0.164·37-s − 0.312·41-s − 0.609·43-s − 0.824·53-s + 1.07·55-s + 1.79·61-s − 1.48·65-s + 1.46·67-s + 0.949·71-s + 1.63·73-s + 1.35·79-s + 0.658·83-s + 0.867·85-s + 0.820·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.349428972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.349428972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.40007661635175, −12.96828211513810, −12.64850388030583, −12.01282453049464, −11.41109059155622, −11.14153403501099, −10.61226992402444, −10.31888824784096, −9.607772958503252, −8.916854846327836, −8.470343961152749, −8.058469003568075, −7.894747206339661, −6.884591170212539, −6.584993911288065, −6.159287172407338, −5.312916907248689, −4.917285080785747, −4.292569420049374, −3.686154372075476, −3.396359046110890, −2.451813593343937, −2.084505042798851, −1.064315075128757, −0.3934363828338999,
0.3934363828338999, 1.064315075128757, 2.084505042798851, 2.451813593343937, 3.396359046110890, 3.686154372075476, 4.292569420049374, 4.917285080785747, 5.312916907248689, 6.159287172407338, 6.584993911288065, 6.884591170212539, 7.894747206339661, 8.058469003568075, 8.470343961152749, 8.916854846327836, 9.607772958503252, 10.31888824784096, 10.61226992402444, 11.14153403501099, 11.41109059155622, 12.01282453049464, 12.64850388030583, 12.96828211513810, 13.40007661635175
