| L(s) = 1 | + 2-s − 2.41·3-s + 4-s + 5-s − 2.41·6-s + 3.41·7-s + 8-s + 2.82·9-s + 10-s + 2·11-s − 2.41·12-s − 5.65·13-s + 3.41·14-s − 2.41·15-s + 16-s − 8.07·17-s + 2.82·18-s − 3·19-s + 20-s − 8.24·21-s + 2·22-s − 2.41·24-s + 25-s − 5.65·26-s + 0.414·27-s + 3.41·28-s + 9.07·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.447·5-s − 0.985·6-s + 1.29·7-s + 0.353·8-s + 0.942·9-s + 0.316·10-s + 0.603·11-s − 0.696·12-s − 1.56·13-s + 0.912·14-s − 0.623·15-s + 0.250·16-s − 1.95·17-s + 0.666·18-s − 0.688·19-s + 0.223·20-s − 1.79·21-s + 0.426·22-s − 0.492·24-s + 0.200·25-s − 1.10·26-s + 0.0797·27-s + 0.645·28-s + 1.68·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162798902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162798902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 8.07T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 29 | \( 1 - 9.07T + 29T^{2} \) |
| 31 | \( 1 - 9.41T + 31T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 - 2.07T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 7.41T + 79T^{2} \) |
| 83 | \( 1 + 4.89T + 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−8.173036857877128173009720136069, −6.98882262869816183711404183654, −6.66713543333082081757058061389, −6.01162108020513210339172361452, −5.05123867212245688029291574403, −4.70020923990107444268882197545, −4.29939478879532896076085707802, −2.62045063049015991091179663335, −1.98164228744792650811789619903, −0.76563286882215492044374834908,
0.76563286882215492044374834908, 1.98164228744792650811789619903, 2.62045063049015991091179663335, 4.29939478879532896076085707802, 4.70020923990107444268882197545, 5.05123867212245688029291574403, 6.01162108020513210339172361452, 6.66713543333082081757058061389, 6.98882262869816183711404183654, 8.173036857877128173009720136069
