| L(s) = 1 | + 1.83·2-s + 2.19·3-s + 1.36·4-s + 0.635·5-s + 4.03·6-s − 1.16·8-s + 1.83·9-s + 1.16·10-s − 11-s + 3.00·12-s + 1.80·13-s + 1.39·15-s − 4.86·16-s + 2.83·17-s + 3.36·18-s + 5.56·19-s + 0.867·20-s − 1.83·22-s + 2.16·23-s − 2.56·24-s − 4.59·25-s + 3.30·26-s − 2.56·27-s − 10.4·29-s + 2.56·30-s − 6.43·31-s − 6.59·32-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 1.26·3-s + 0.682·4-s + 0.284·5-s + 1.64·6-s − 0.412·8-s + 0.611·9-s + 0.368·10-s − 0.301·11-s + 0.866·12-s + 0.499·13-s + 0.360·15-s − 1.21·16-s + 0.687·17-s + 0.793·18-s + 1.27·19-s + 0.193·20-s − 0.391·22-s + 0.451·23-s − 0.523·24-s − 0.919·25-s + 0.647·26-s − 0.493·27-s − 1.93·29-s + 0.467·30-s − 1.15·31-s − 1.16·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.846341323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.846341323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 - 0.635T + 5T^{2} \) |
| 13 | \( 1 - 1.80T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 19 | \( 1 - 5.56T + 19T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 4.33T + 61T^{2} \) |
| 67 | \( 1 + 1.60T + 67T^{2} \) |
| 71 | \( 1 - 4.29T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 4.76T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 + 0.364T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−11.05476462865789793855992402061, −9.645785422192813832005404663710, −9.182004117447016647764703533422, −8.061907291853832428313120504974, −7.23706135762800034880575366888, −5.86936232016234666188234429215, −5.20254592300022260873508919793, −3.75693777485538633556633716211, −3.28052710639986430591421548126, −2.05077424246436447289943156546,
2.05077424246436447289943156546, 3.28052710639986430591421548126, 3.75693777485538633556633716211, 5.20254592300022260873508919793, 5.86936232016234666188234429215, 7.23706135762800034880575366888, 8.061907291853832428313120504974, 9.182004117447016647764703533422, 9.645785422192813832005404663710, 11.05476462865789793855992402061
