| L(s) = 1 | − 2-s + 4-s + 1.40·5-s + 5.24·7-s − 8-s − 1.40·10-s − 11-s − 0.638·13-s − 5.24·14-s + 16-s − 3.31·17-s + 19-s + 1.40·20-s + 22-s − 2.63·23-s − 3.03·25-s + 0.638·26-s + 5.24·28-s − 1.23·29-s − 3.15·31-s − 32-s + 3.31·34-s + 7.34·35-s + 11.9·37-s − 38-s − 1.40·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.626·5-s + 1.98·7-s − 0.353·8-s − 0.442·10-s − 0.301·11-s − 0.177·13-s − 1.40·14-s + 0.250·16-s − 0.802·17-s + 0.229·19-s + 0.313·20-s + 0.213·22-s − 0.550·23-s − 0.607·25-s + 0.125·26-s + 0.991·28-s − 0.229·29-s − 0.567·31-s − 0.176·32-s + 0.567·34-s + 1.24·35-s + 1.96·37-s − 0.162·38-s − 0.221·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930474958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930474958\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 - 5.24T + 7T^{2} \) |
| 13 | \( 1 + 0.638T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 23 | \( 1 + 2.63T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 + 2.63T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 + 8.25T + 61T^{2} \) |
| 67 | \( 1 - 8.11T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.52T + 97T^{2} \) |
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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
−8.467027461817242078035064872419, −7.73177975248227367106388727650, −7.46583523338367062395817849056, −6.21688587155681447962753919501, −5.61524998561701430702922619837, −4.76837350052846110070976906046, −4.03262311958941014920451736472, −2.42938394552470638746927846086, −2.00746907003076041945660637819, −0.931862569391702301840175373842,
0.931862569391702301840175373842, 2.00746907003076041945660637819, 2.42938394552470638746927846086, 4.03262311958941014920451736472, 4.76837350052846110070976906046, 5.61524998561701430702922619837, 6.21688587155681447962753919501, 7.46583523338367062395817849056, 7.73177975248227367106388727650, 8.467027461817242078035064872419
