| L(s) = 1 | + 2.56·2-s − 1.40·4-s − 7·7-s − 24.1·8-s − 66.4·11-s + 34.0·13-s − 17.9·14-s − 50.7·16-s + 6.12·17-s − 163.·19-s − 170.·22-s + 35.8·23-s + 87.4·26-s + 9.82·28-s − 27.7·29-s + 74.3·31-s + 62.7·32-s + 15.7·34-s + 260.·37-s − 419.·38-s + 445.·41-s + 474.·43-s + 93.2·44-s + 92.1·46-s − 51.0·47-s + 49·49-s − 47.7·52-s + ⋯ |
| L(s) = 1 | + 0.908·2-s − 0.175·4-s − 0.377·7-s − 1.06·8-s − 1.82·11-s + 0.726·13-s − 0.343·14-s − 0.793·16-s + 0.0874·17-s − 1.97·19-s − 1.65·22-s + 0.325·23-s + 0.659·26-s + 0.0663·28-s − 0.177·29-s + 0.430·31-s + 0.346·32-s + 0.0794·34-s + 1.15·37-s − 1.79·38-s + 1.69·41-s + 1.68·43-s + 0.319·44-s + 0.295·46-s − 0.158·47-s + 0.142·49-s − 0.127·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.621196417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621196417\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 2.56T + 8T^{2} \) |
| 11 | \( 1 + 66.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.12T + 4.91e3T^{2} \) |
| 19 | \( 1 + 163.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 74.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 260.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 445.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 474.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 51.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 115.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 390.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 713.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 810.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 350.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 50.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 84.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.23e3T + 9.12e5T^{2} \) |
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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
−8.985288648269438469916067193096, −8.284358679395530470120293364348, −7.46984132420170460316377385332, −6.15846617002329424639599122002, −5.89218487013013100882931600828, −4.75426496463200490383419383780, −4.18932225213497655938074947140, −3.06590117663495137972127366588, −2.35234677929613340495336488414, −0.50990983025790757677537338363,
0.50990983025790757677537338363, 2.35234677929613340495336488414, 3.06590117663495137972127366588, 4.18932225213497655938074947140, 4.75426496463200490383419383780, 5.89218487013013100882931600828, 6.15846617002329424639599122002, 7.46984132420170460316377385332, 8.284358679395530470120293364348, 8.985288648269438469916067193096
