| L(s) = 1 | + 1.21·2-s − 0.534·4-s + 2.21·5-s − 3.06·8-s + 2.67·10-s + 0.789·11-s + 13-s − 2.64·16-s − 1.74·17-s − 4.32·19-s − 1.18·20-s + 0.955·22-s + 1.11·23-s − 0.112·25-s + 1.21·26-s + 8.48·29-s − 5.70·31-s + 2.93·32-s − 2.11·34-s + 2.27·37-s − 5.23·38-s − 6.78·40-s + 12.1·41-s + 8.06·43-s − 0.421·44-s + 1.34·46-s + 8.74·47-s + ⋯ |
| L(s) = 1 | + 0.856·2-s − 0.267·4-s + 0.988·5-s − 1.08·8-s + 0.846·10-s + 0.237·11-s + 0.277·13-s − 0.661·16-s − 0.423·17-s − 0.991·19-s − 0.264·20-s + 0.203·22-s + 0.231·23-s − 0.0225·25-s + 0.237·26-s + 1.57·29-s − 1.02·31-s + 0.518·32-s − 0.362·34-s + 0.374·37-s − 0.849·38-s − 1.07·40-s + 1.89·41-s + 1.23·43-s − 0.0635·44-s + 0.198·46-s + 1.27·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.047051106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.047051106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 5 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 - 0.789T + 11T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 - 1.11T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 6.55T + 67T^{2} \) |
| 71 | \( 1 + 5.85T + 71T^{2} \) |
| 73 | \( 1 + 8.00T + 73T^{2} \) |
| 79 | \( 1 + 6.91T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 - 3.39T + 89T^{2} \) |
| 97 | \( 1 - 0.0981T + 97T^{2} \) |
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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
−8.261746835962733837782609910487, −7.20039365796126556316186932878, −6.35365107260189317371696072967, −5.93370325713693477536835606314, −5.28559146129598433197728802302, −4.37959064457646464031837185132, −3.93971443340380005508017874056, −2.79033214008178647868612203736, −2.15795606356188588502367455236, −0.816931932107567233151666476934,
0.816931932107567233151666476934, 2.15795606356188588502367455236, 2.79033214008178647868612203736, 3.93971443340380005508017874056, 4.37959064457646464031837185132, 5.28559146129598433197728802302, 5.93370325713693477536835606314, 6.35365107260189317371696072967, 7.20039365796126556316186932878, 8.261746835962733837782609910487
