| L(s) = 1 | + 2-s + 4-s + 1.11·5-s − 1.85·7-s + 8-s + 1.11·10-s + 0.749·11-s + 4.65·13-s − 1.85·14-s + 16-s + 2.53·17-s + 2.66·19-s + 1.11·20-s + 0.749·22-s − 23-s − 3.75·25-s + 4.65·26-s − 1.85·28-s + 4.55·29-s + 1.05·31-s + 32-s + 2.53·34-s − 2.07·35-s + 0.587·37-s + 2.66·38-s + 1.11·40-s − 6.70·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.499·5-s − 0.700·7-s + 0.353·8-s + 0.353·10-s + 0.225·11-s + 1.29·13-s − 0.495·14-s + 0.250·16-s + 0.613·17-s + 0.611·19-s + 0.249·20-s + 0.159·22-s − 0.208·23-s − 0.750·25-s + 0.912·26-s − 0.350·28-s + 0.845·29-s + 0.189·31-s + 0.176·32-s + 0.434·34-s − 0.350·35-s + 0.0965·37-s + 0.432·38-s + 0.176·40-s − 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.465497294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.465497294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.11T + 5T^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 11 | \( 1 - 0.749T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 - 1.05T + 31T^{2} \) |
| 37 | \( 1 - 0.587T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 + 0.967T + 61T^{2} \) |
| 67 | \( 1 - 7.55T + 67T^{2} \) |
| 71 | \( 1 - 9.39T + 71T^{2} \) |
| 73 | \( 1 - 3.04T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 9.80T + 83T^{2} \) |
| 89 | \( 1 + 9.50T + 89T^{2} \) |
| 97 | \( 1 - 9.98T + 97T^{2} \) |
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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
−8.449243713041258398674845027432, −7.73112864466965469019357134575, −6.74883482211058226231019152289, −6.20151387866595098217533239701, −5.64599975499144424408764798810, −4.76030535192769795772044455073, −3.70280519989163334910465633373, −3.25113685950294309663373068136, −2.11814608056058195287874425539, −1.02788096284788352859712231296,
1.02788096284788352859712231296, 2.11814608056058195287874425539, 3.25113685950294309663373068136, 3.70280519989163334910465633373, 4.76030535192769795772044455073, 5.64599975499144424408764798810, 6.20151387866595098217533239701, 6.74883482211058226231019152289, 7.73112864466965469019357134575, 8.449243713041258398674845027432
