| L(s) = 1 | − 2-s − 3.22·3-s + 4-s + 1.50·5-s + 3.22·6-s + 7-s − 8-s + 7.41·9-s − 1.50·10-s + 2.69·11-s − 3.22·12-s + 2.34·13-s − 14-s − 4.86·15-s + 16-s − 1.99·17-s − 7.41·18-s − 3.26·19-s + 1.50·20-s − 3.22·21-s − 2.69·22-s + 5.95·23-s + 3.22·24-s − 2.72·25-s − 2.34·26-s − 14.2·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.86·3-s + 0.5·4-s + 0.674·5-s + 1.31·6-s + 0.377·7-s − 0.353·8-s + 2.47·9-s − 0.477·10-s + 0.811·11-s − 0.931·12-s + 0.650·13-s − 0.267·14-s − 1.25·15-s + 0.250·16-s − 0.482·17-s − 1.74·18-s − 0.749·19-s + 0.337·20-s − 0.704·21-s − 0.573·22-s + 1.24·23-s + 0.658·24-s − 0.544·25-s − 0.459·26-s − 2.74·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9928238315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9928238315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 3.22T + 3T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 + 1.99T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 9.75T + 31T^{2} \) |
| 37 | \( 1 - 2.88T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 + 3.37T + 53T^{2} \) |
| 59 | \( 1 + 0.771T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2.71T + 79T^{2} \) |
| 83 | \( 1 + 7.89T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.000394036467480572191861098736, −7.11854221958254349268184256024, −6.57677368552912846972554531589, −5.99191040853449798324651932542, −5.53519746689985528841471620642, −4.55804026540525024535494961465, −3.95125387622400902457625108958, −2.36773049022173897414277098668, −1.40428054177587705705312919568, −0.72034051256466015053201436295,
0.72034051256466015053201436295, 1.40428054177587705705312919568, 2.36773049022173897414277098668, 3.95125387622400902457625108958, 4.55804026540525024535494961465, 5.53519746689985528841471620642, 5.99191040853449798324651932542, 6.57677368552912846972554531589, 7.11854221958254349268184256024, 8.000394036467480572191861098736
