| L(s) = 1 | + 3.27·3-s + 0.447·7-s + 7.69·9-s − 2.50·11-s − 3.23·13-s + 6.79·17-s − 6.54·19-s + 1.46·21-s + 7.94·23-s + 15.3·27-s + 6.66·29-s + 3.96·31-s − 8.18·33-s + 37-s − 10.5·39-s + 5.66·41-s − 5.28·43-s + 7.75·47-s − 6.79·49-s + 22.2·51-s + 4.46·53-s − 21.4·57-s + 9.32·59-s − 12.2·61-s + 3.44·63-s + 1.82·67-s + 25.9·69-s + ⋯ |
| L(s) = 1 | + 1.88·3-s + 0.169·7-s + 2.56·9-s − 0.754·11-s − 0.896·13-s + 1.64·17-s − 1.50·19-s + 0.319·21-s + 1.65·23-s + 2.95·27-s + 1.23·29-s + 0.711·31-s − 1.42·33-s + 0.164·37-s − 1.69·39-s + 0.884·41-s − 0.805·43-s + 1.13·47-s − 0.971·49-s + 3.11·51-s + 0.613·53-s − 2.83·57-s + 1.21·59-s − 1.57·61-s + 0.434·63-s + 0.222·67-s + 3.12·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.587740189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.587740189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 3.27T + 3T^{2} \) |
| 7 | \( 1 - 0.447T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 + 6.54T + 19T^{2} \) |
| 23 | \( 1 - 7.94T + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 - 4.46T + 53T^{2} \) |
| 59 | \( 1 - 9.32T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 1.82T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 1.04T + 79T^{2} \) |
| 83 | \( 1 + 8.59T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.34T + 97T^{2} \) |
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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
−8.044506079235692527048631655202, −7.35041764518115123248936761450, −6.91656389008049595352959990643, −5.77944304712094876676754490218, −4.73956642852383867110524656694, −4.34232655924232755951551152009, −3.15944039825729413298451347197, −2.86756513196427945019633809359, −2.06967643671791084909932689054, −1.02409608096304957163464513922,
1.02409608096304957163464513922, 2.06967643671791084909932689054, 2.86756513196427945019633809359, 3.15944039825729413298451347197, 4.34232655924232755951551152009, 4.73956642852383867110524656694, 5.77944304712094876676754490218, 6.91656389008049595352959990643, 7.35041764518115123248936761450, 8.044506079235692527048631655202
