| L(s) = 1 | − 1.69·2-s − 3-s + 0.868·4-s − 1.90·5-s + 1.69·6-s + 1.91·8-s + 9-s + 3.22·10-s − 5.19·11-s − 0.868·12-s + 2.94·13-s + 1.90·15-s − 4.98·16-s − 4.99·17-s − 1.69·18-s + 0.972·19-s − 1.65·20-s + 8.79·22-s + 23-s − 1.91·24-s − 1.38·25-s − 4.99·26-s − 27-s − 4.08·29-s − 3.22·30-s + 1.11·31-s + 4.60·32-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 0.577·3-s + 0.434·4-s − 0.850·5-s + 0.691·6-s + 0.677·8-s + 0.333·9-s + 1.01·10-s − 1.56·11-s − 0.250·12-s + 0.817·13-s + 0.491·15-s − 1.24·16-s − 1.21·17-s − 0.399·18-s + 0.223·19-s − 0.369·20-s + 1.87·22-s + 0.208·23-s − 0.391·24-s − 0.276·25-s − 0.978·26-s − 0.192·27-s − 0.757·29-s − 0.588·30-s + 0.199·31-s + 0.814·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2322591759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2322591759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 5 | \( 1 + 1.90T + 5T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 + 4.99T + 17T^{2} \) |
| 19 | \( 1 - 0.972T + 19T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 + 9.85T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 5.61T + 71T^{2} \) |
| 73 | \( 1 - 6.42T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 2.34T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 0.186T + 97T^{2} \) |
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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
−8.541272445953434369147343830104, −7.895318528749526472704257653322, −7.46899688497941502349741095820, −6.59431495467350484678949516673, −5.66086126107053125128722694535, −4.72353898547855598634804386811, −4.09949764774211974164668708288, −2.84875741500893840310319653811, −1.65004463048718612823718520204, −0.35456173980449114931987721187,
0.35456173980449114931987721187, 1.65004463048718612823718520204, 2.84875741500893840310319653811, 4.09949764774211974164668708288, 4.72353898547855598634804386811, 5.66086126107053125128722694535, 6.59431495467350484678949516673, 7.46899688497941502349741095820, 7.895318528749526472704257653322, 8.541272445953434369147343830104
