| L(s) = 1 | + 0.222·2-s + 3-s − 1.95·4-s − 0.636·5-s + 0.222·6-s − 1.72·7-s − 0.877·8-s + 9-s − 0.141·10-s + 4.95·11-s − 1.95·12-s + 2.50·13-s − 0.384·14-s − 0.636·15-s + 3.70·16-s + 0.222·18-s − 0.950·19-s + 1.24·20-s − 1.72·21-s + 1.09·22-s − 2.12·23-s − 0.877·24-s − 4.59·25-s + 0.556·26-s + 27-s + 3.37·28-s + 9.58·29-s + ⋯ |
| L(s) = 1 | + 0.157·2-s + 0.577·3-s − 0.975·4-s − 0.284·5-s + 0.0907·6-s − 0.653·7-s − 0.310·8-s + 0.333·9-s − 0.0447·10-s + 1.49·11-s − 0.563·12-s + 0.695·13-s − 0.102·14-s − 0.164·15-s + 0.926·16-s + 0.0523·18-s − 0.218·19-s + 0.277·20-s − 0.377·21-s + 0.234·22-s − 0.442·23-s − 0.179·24-s − 0.918·25-s + 0.109·26-s + 0.192·27-s + 0.637·28-s + 1.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.601849450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.601849450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.222T + 2T^{2} \) |
| 5 | \( 1 + 0.636T + 5T^{2} \) |
| 7 | \( 1 + 1.72T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 - 2.50T + 13T^{2} \) |
| 19 | \( 1 + 0.950T + 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 - 9.58T + 29T^{2} \) |
| 31 | \( 1 - 5.27T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 + 6.44T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 2.96T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 + 0.384T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 + 2.37T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 1.98T + 89T^{2} \) |
| 97 | \( 1 - 3.04T + 97T^{2} \) |
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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
−9.764464811886636511142975033575, −9.396628236284825040726095301219, −8.529264463988429356438407167125, −7.87761658656161682931531229925, −6.56691151042302356558644670277, −5.93932232992331387427846798535, −4.35454511804715088539774576545, −3.98666856142688385078573779875, −2.88260412275820003463458089772, −1.03989225069973365425781144792,
1.03989225069973365425781144792, 2.88260412275820003463458089772, 3.98666856142688385078573779875, 4.35454511804715088539774576545, 5.93932232992331387427846798535, 6.56691151042302356558644670277, 7.87761658656161682931531229925, 8.529264463988429356438407167125, 9.396628236284825040726095301219, 9.764464811886636511142975033575
