| L(s) = 1 | + 4.13·2-s + 8.07·3-s + 9.07·4-s − 5·5-s + 33.3·6-s + 3.79·7-s + 4.42·8-s + 38.1·9-s − 20.6·10-s + 34.0·11-s + 73.2·12-s + 41.7·13-s + 15.6·14-s − 40.3·15-s − 54.2·16-s + 157.·18-s + 58.8·19-s − 45.3·20-s + 30.6·21-s + 140.·22-s + 98.6·23-s + 35.6·24-s + 25·25-s + 172.·26-s + 89.9·27-s + 34.4·28-s + 208.·29-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 1.55·3-s + 1.13·4-s − 0.447·5-s + 2.26·6-s + 0.205·7-s + 0.195·8-s + 1.41·9-s − 0.653·10-s + 0.933·11-s + 1.76·12-s + 0.891·13-s + 0.299·14-s − 0.694·15-s − 0.848·16-s + 2.06·18-s + 0.709·19-s − 0.507·20-s + 0.318·21-s + 1.36·22-s + 0.894·23-s + 0.303·24-s + 0.200·25-s + 1.30·26-s + 0.641·27-s + 0.232·28-s + 1.33·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.247383345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.247383345\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 4.13T + 8T^{2} \) |
| 3 | \( 1 - 8.07T + 27T^{2} \) |
| 7 | \( 1 - 3.79T + 343T^{2} \) |
| 11 | \( 1 - 34.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 58.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 208.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 288.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 317.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 705.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 78.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 32.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 571.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 455.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 463.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 794.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.45e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 606.T + 9.12e5T^{2} \) |
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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
−8.904585388914063090622103238529, −8.493254560600903518139404980501, −7.40411119393260734682546125433, −6.74769389410829546384133383383, −5.69696502921624616358892932189, −4.63922710512167433531948222648, −3.82709224345710291014396915372, −3.36514877090492704981077074053, −2.48359745429594197218572258500, −1.25226681015453440170635485980,
1.25226681015453440170635485980, 2.48359745429594197218572258500, 3.36514877090492704981077074053, 3.82709224345710291014396915372, 4.63922710512167433531948222648, 5.69696502921624616358892932189, 6.74769389410829546384133383383, 7.40411119393260734682546125433, 8.493254560600903518139404980501, 8.904585388914063090622103238529
